Download EQUATIONS and INEQUALITIES

Equations and Inequalities
Rational Equations
Overview of Objectives, students should be able to:
1. Solve rational equations with variables in the
denominators.
2. Recognize identities, conditional equations, and
inconsistent equations.
Objectives:
Solve rational equations
Main Overarching Questions:
1. How do you solve rational equations?
2. What is an extraneous solution?
Activities and Questions to ask students:
Give students a simple example of a rational equation:
1
2
x
3
.
x
Ask students to observe what value of x is “not allowed.” If they are not sure, says we are
never allowed to have ‘blank’ in the denominator.”
Next ask them how they might proceed to solve the equation. Why is this equation NOT
linear? (It has fractions)
How can we clear our fractions? What one quantity can we multiply by to ensure all fractions
are cleared?
Ask students what the LCD is in the previous example
1
2
x
3
.
x
Do we have to multiply the 2 by the LCD? Why?
Have students multiply through by LCD to clear the fraction. What kind of equation is left
over? How do we solve it?
Have students summarize the process of solving radical equations.
Give students another rational equation:
Determine whether an equation represents an
identity, conditional, or inconsistent statement.
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x
3
x 3
x 3
9 . Ask them to solve it.
What solution did you get? Does this solution work when plugged in? How could we have
known the solution would not work?
Define the solution x = 3 in the previous example as an extraneous solution. Remind
students to always list out restrictions on the variable at the beginning of the problem.
Since this is a MAC 1105 topic, make sure to graduate to more difficult problems that involve
multiple factors in the LCD.
Give students a simple identity: x 2 x 2 for example. Ask them what they notice?
What is the solution to this equation?
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Some students make call out specific values of x. Take note of them and continue to ask if
there are other solutions. How many solutions are there?
Once the students see that every real number is a solution to this equation, ask them how
they could tell this might happen from looking at the equation. What would happen if we
had tried to solve for x? What would have been left over?
Have students see that solving for x, would have resulted in the true statement 2 2 . Point
out that equations of this type are called identities. Identities have the formal solution
{x | x is a real number}
Give students another identity: 4( x 7) 2(2 x 14) . Ask them to solve the equation.
What happens? What type of equation is this? What is its solutions?
Next, give students a simple inconsistent equation: x 2 x 3 . Ask them what they
notice? What is the solution to this equation?
If students call out specific values of x, make sure to have the class check them.
Once the students see that no real number will satisfy the equation, ask them how they
could tell this might happen from looking at the equation. What would happen if we had
tried to solve for x? What would have been left over?
Have students see that solving for x, would have resulted in the false statement 2 3 . Point
out that equations of this type are called inconsistencies. Inconsistencies have no real
solution.
Give students another identity: 8x 2 x 10 x 6 . Ask them to solve the equation. What
happens? What type of equation is this? What is its solutions?
Point out that all other equations that we have solved resulting in at least one solution (but
not all real numbers as the solution set) are called conditional equations.
Complex Numbers
Overview of Objectives, students should be able to:
Main Overarching Questions:
1. Define complex numbers.
2. Add and subtract complex numbers
3. Multiply complex numbers.
4. Divide complex numbers.
5. Perform operations with square roots of negative
numbers.
1. What is a complex number?
2. How do we perform operations on complex numbers?
Objectives:
Define complex numbers
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Activities and Questions to ask students:
Begin by introducing
4 . Ask students what it is. What does it simplify to? If students say
2 or -2, have all students verify this is not correct.
2
Once students see that there is no REAL number to represent
unit i
1 . What is i ?
Ask students how we might now rewrite
Add and subtract complex numbers.
Multiply complex numbers.
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4 in terms of i. If students have trouble,
1
4
1
suggest the factorization:
4
4
Have students practice with a few other imaginary roots to attain mastery.
Next, state the definition of a complex number: a bi , where a and b are real numbers.
Give students an example like 3 4 i Tell students the real part is 3 and the imaginary part is
4.
To extend the concept, ask students if the real number 5 is in fact a complex number. How
could we rewrite it in the complex form? What is the real part? What is the imaginary part?
What about the purely imaginary number 4i? Is it complex? What is the real part? What is
the imaginary part?
The idea is to get students to see they are simply combining like terms. Have students
simplify the linear expression: (2 3x) (4 5 x) . How did you simplify?
Have them repeat the process with (2 3i ) (4 5i )
Make sure to tell students that complex numbers must be in a bi form to be completely
simplified.
How about subtraction? (3 7i ) (3 2i)
How about the problem: 2(5 i ) 3(2 4i ) . If students are unsure, tell them to consider
how they would proceed if we substituted x for i.
Again, the idea is comparable to multiplying polynomials. Start with the simple distribution:
2 x( x 3) . How would you simplify this?
Now, what about 2i(i
Divide complex numbers
4 , define the imaginary
2
3) ? Most students should see that 2i(i 3)
2i 2
6i .
But this isn’t in simplified form, what is i 2 ? Have students simplify the rest of the problem.
Give students a FOIL problem: ( 5 4i )(3 i) . Ask students what previously learned
multiplication rule will come in handy. If a review of FOIL is necessary, take a minute to
review the steps, then continue. If students forget to convert i 2 , remind them their answer
needs to be in the form a bi .
For further practice and review give another example to try: (2 3i ) 2 . Follow the same line
of questioning as did with FOIL.
If we want to divide two complex numbers, we seek to remove the complex number in the
denominator. But how do we get rid of this complex number?
Ask students to recall how they rationalized the denominator in their previous math
3
class:
3
2
examples:
x
? What did we have to multiply by? What would we need to multiply by in the
2
2 i
For another approach, have students multiply (a bi)(a bi) . What is the result? What
type of number is this? Tell students that we are multiplying by what is called the complex
conjugate.
What is the complex conjugate of 2 i ?
Next, can we just multiply the denominator by the conjugate 2 i ? Why not? How do we
keep the original fraction equivalent (balanced)? Have students see that we need to multiply
the numerator AND denominator by the complex conjugate. Have students finish the
problem.
Give students another example to practice of similar type.
Perform operations with square roots of negative
numbers.
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16 . How do we simplify this
To get the discussion going, give the simple example
expression? If students try to ADD the radicands, remind them of their radical rules. You can
also have students demonstrate on the calculator this will not work.
What did we need to do first? Have students note that they must simplify the imaginary
radical FIRST, before combining together.
64
25 ? Again, if any student tries to “cancel” the two negatives,
How about
remind them of radical rules and use the calculator to demonstrate the error in their logic.
Give a slightly more complicated example like 5
different? How do we proceed?
16
3
81 . How is this example
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4 . How do we simplify? A logical (yet
Next, give a multiplication problem:
incorrect) way to proceed, is to combine the two radicals via multiplication. If students
follow this route, explain that the multiplication rule does not apply for imaginary numbers.
How did we begin our simplification of the last two problems? CONVERT, then MULTIPLY.
Have students compare the answers by the incorrect and correct method, if the former
answer was previously found.
Quadratic Equations
Overview of Objectives, students should be able to:
Main Overarching Questions:
1. Solve quadratic equations by factoring.
2. Solve quadratic equations by the square root
property.
1. How do we solve quadratic equations?
2. How do we decide which method is best to solve a particular quadratic equation?
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3. Solve quadratic equations by completing the square.
4. Solve quadratic equations using the quadratic
formula.
5. Use the discriminant to determine the number and
type of solutions.
6. Determine the most efficient method to use when
solving a quadratic equation.
Objectives:
Activities and Questions to ask students:
Solve quadratic equations by factoring.
Give students the general quadratic equation ax 2 bx c 0
How do you know an equation is quadratic? What properties does it have?
Give students a simple product such as 4 0 and ask for the result.
Then reverse the order and ask for the result of 0 4 .
Ask the students if they notice any similarities between the two simple expressions.
Why is the final result the same in each case?
What requirement must be met for the product of two numbers to be 0?
Next, give the students the simple equation: a b 0 . What are the possible solutions to
this equation? Is there more than one solution?
Have students establish the zero product property: if the product of two factors is 0, then
either factor is 0.
Give students a very simple quadratic equation to solve like ( x 2)( 2 x 3) 0
How would we solve this equation? If necessary, remind students of what has been
discussed so far this discussion.
Next give students a simple quadratic equation: x 2 4 x 0
Ask students how they might attempt to solve the equation.
How can we use the zero product property to aid in solving the equation?
If we need a product to use the property, how can we transform our sum of terms into a
product? If students have trouble, give students an arithmetic example to illustrate the
point: “How would we write 10 as a product?”
Then, have students solve x 2 4 x 0 . What steps did you use to solve the equation?
Give students another equation to solve using a different factoring method like
x2
4x 3
0
What differences did you notice in solving this equation?
Give students one additional equation that does not have 0 on the right hand side:
2
x 3x 3 1 . What additional steps might be necessary? Summarize the process.
In each example, how many solutions did we get? Is there a relationship between the degree
of the equation and the number of solutions?
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Solve quadratic equations by the square root
property.
Give students a very simple quadratic equation to solve like: x 2 4 .
If students only give x = 2 as the solution ask them if there any other solutions. If necessary,
ask them how many solutions we normally have when solving quadratic equation.
Next, ask students (this has already been discussed) what x 2 is.
How can we “get rid of” a square on a variable?
Now, going back to x 2 4 ask the students what other process we might use to solve the
equation.
If we take the square root of both sides, what additional steps do we need to get both
solutions?
Have students summarize the process of solving quadratic equations by taking square roots.
Give students another quadratic equation to solve: 2 x 2 18 . What is different about this
equation? What step could we use to make the equation look similar to the first one?
Have students summarize the steps to solve this equation.
What steps would be necessary to solve: 5 x 2 2 27 ? What is different about this
equation? What step or steps would be necessary to make it look like the first equation?
2
Solve quadratic equations by completing the square.
Finally, have students summarize the steps to solve x 2
4 5 . What is different
about the “squared” portion of this equation?
After taking the square root of both sides what additional step or steps is necessary?
Begin by explaining the purpose of the method. We want to add a constant to the x 2 bx
2
to obtain a perfect square. Say for example we want to solve x 8x 9
First give a few simple examples: x 2 8x ____ (x 4)2 . Have students FOIL the right
hand side to confirm. Repeat with a few other simple examples.
In each case what did we need to add (i.e. fill in the blank)? Is there a pattern?
Have students draw the conclusion that generally
b
is added to complete the square.
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Now, return to the problem at hand. If we need to complete the square on the left hand side
for x 2 8x 9 , we need to add 16. But if we add 16 to the LHS, what do we need to do to
the RHS? (add 16). Have students do this.
So we have
Solve quadratic equations using the quadratic
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x 2 8x 16
(x 4)
2
9 16
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Now, how do we proceed to solve? Have students use the square root property to finish the
problem.
Have students summarize the process of completing the square.
Give students the general quadratic equation and quadratic formula: ax 2 bx c 0 and
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formula.
x
Use the discriminant to determine the number and
type of solutions.
b2
2a
b
4ac
.
What do you notice about the formula? What does the formula give us? How many
solutions should we get? In what instances would we get 1 solution? No solutions?
Ask students how could they identify what the a, b, and c values are.
Give students the example 3x 2 4 x 1 0 . Ask students to summarize the process they
use to solve the equation. Ask in particular that they summarize the steps in simplifying the
expression.
What about 2 x 2 4 x 1 ? What are the a, b, and c values? What additional step do we
need before plugging into the formula?
First, ask students how many solutions we have usually encountered when solving quadratic
equations.
Review the quadratic formula. Looking at the formula, how does it give us two solutions?
(The + and -). If some students do not see this, ask them what solutions are yielded from
x 2 4.
What happens AFTER the ? (The square root)
Now, ask students if there is way to only get one solution. To draw out this concept, ask
them what number you can add and subtract and still have the SAME number. (Zero).
So, what happens when the inside of the square root is 0? How many solutions do we have?
Give a simple example if necessary.
Next, ask them what happens if the inside of the square root is negative? What kind of
values do we get?
Summarize the results, with 3 “plugged in” quadratic formula problems:
Quadratic Formula
Number of Solutions
Type of Solutions
4
16
2
4
0
2
4
16
2
In all cases, what value controls the number and type of solution? Have students see the
inside of the square root solely determines the number and type of solutions. Define the
inside of the square root from the quadratic formula: b 2 4ac as the disriminant.
Have students see that they only need to calculate this quantity to see what type of solutions
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and how many there are to the quadratic formula.
What happens if the discriminant is positive?
Zero?
Negative?
Give students several examples to work.
Determine the most efficient method to use when
solving a quadratic equation.
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Give students several quadratic equations, one of which is factorable, another that is in
u 2 d form, and a third that is not factorable or in the square root property form. Ask
students what method or methods they COULD use to solve each equation.
Then ask students which method they would PREFER to use.
Ask students which method can be used to solve ANY quadratic equation.
Students should realize they can always use quadratic formula.
Ask students which method is the easiest. Hopefully they realize that checking for factoring
is normally the best first step, but some will prefer to use the quadratic formula every time.
Ask students how they might spot a square root property type of problem.
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Other Types of Equations
Overview of Objectives, students should be able to:
Main Overarching Questions:
1. Solve polynomial equations by factoring.
2. Solve radical equations.
3. Solve equations with rational exponents.
4. Solve equations using substitutions.
1.
2.
3.
4.
5.
5. Solve absolute value equations.
Objectives:
Solve polynomial equations by factoring
Solve radical equations
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How do we solve polynomial equations using factoring?
How do we solve radical equations?
How do we solve equations with rational exponents?
How do we solve equations using substitutions?
How do we solve absolute value equations?
Activities and Questions to ask students:
Give students a simple polynomial equation like: 3x 4 48 x 2 0
How is this equation different from the quadratic equations studied earlier?
Ask students to think about the easiest way to solve quadratic equations. What did
we do? Have students come to the conclusion that factoring can sometimes be used
to solve polynomial equations.
Have students factor the left hand side: 3x 2 ( x 4)( x 4) 0
Ask students how we proceeded before. Hopefully, they remember that we can use
the zero product property to set each factor to zero:
3x 2 0 or x 4 0 or x 4 0
Have students solve and check their solutions.
Give another example, but in non-general form: 5 x 4 20 x 2 . If students want to
divide by x 2 , remind them that we never divide by a factor of x, since x could possibly
be zero and we can never divide by zero.
How can we make it look like the first example? Once students subtract the right
hand side, remind them this is the general form. Have students solve and check.
Give one more example that requires a different factoring technique like grouping:
x3 x 2 4 x 4 . Have students solve and check.
As an aside, ask students to count up the number of solutions in each example
worked. How many were there? What was the most number of solutions we could
have with quadratic equations? Does it have anything to do with the degree of the
equation? Can we form a pattern? What is it?
Give students a simple example: if x = 2. What is x2 = ? What about if x = 3, x2 = ?
Do you see a pattern? What “operation” are we performing on both sides of the equation?
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Have students draw the conclusion that if a
b then a 2 b 2 (squaring principle)
x 4 . If students just observe the answer is 16,
Now ask students how they would solve:
ask them how they would solve the equation using the squaring principle. Does it match the
solution you observed?
How can you check your solution is correct? Write down the process you used to solve and
check your answer.
4 ? What happens if you use the squaring principle? How could
How do you solve x
you check that 16 is not the solution? Mention that solutions that do not work in an
equation but that are the result of an algebraic method are called “extraneous” solutions
Give another example similar to this one. Do you see a pattern? How could you predict
there would be no solution?
2 . How is this example different than the last one?
How would you solve: x 4 6
How would you need to modify your process to solve it? If students take exception to the -2
on the right hand side, ask them if the radical has been isolated. Remind them that in the
previous simple example, the radical was by itself when we realized the negative on the right
hand side would give an extraneous solution.
2
x 4 2 . Have you done this before? Which
Ask students how they would find:
previously learned processes or rules are you using?
Solve equations with rational exponents.
x 4 2 ? What is different about this example than the
How would you solve: x
last ones? How can you use the squaring process at the beginning to help you solve?
After using the squaring property you still have a radical in the equation, now what do you
do? How many radicals do you have now? Does it look similar to the first type of radical
equations you solved? Write the process you would use to continue.
Ask students to summarize the process of solving radical equations of the types studied.
Give students a worksheet with several radical equations (1 and 2 radicals, some with real
solutions, and some with no real solution) to complete. Have them use the process they wrote
down.
Is there more than one way to solve? Compare with your classmates.
A brief review of rational exponents might be necessary.
First, try to get students to understand why we raise both sides of the equation to the
reciprocal of the given power. Start with a simple example: How we would solve x
1
2
4 . If
1
students have trouble, ask them if there is another way to write x 2 . Hopefully students see
the squaring both sides is on order. What is the solution? (16). Make sure students see that
16 is the ONLY solution.
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Now, how about x
1
3
2 ? What do we need to do to both sides?
How are 3 and 1 related? How are 2 and 1 related?
3
2
Now, have students try x
2
3
4 . They should see what to do, but they may have trouble
3
Solve equations using substitutions.
Solve absolute value equations.
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evaluating x 4 2 . Have them use the definition of rational exponents to calculate this
value and use the calculator as a check.
Was 8 the only solution to the previous equation? Would -8 have also worked? Have
students show by substitution that both 8 and -8 are solutions.
Repeat with another rational exponent with an even numerator.
Now, follow the same steps with an exponent with an odd numerator. Do both the positive
and negative solutions work?
Have students draw the conclusion that if the power has an even numerator, we keep both
the positive and negative solutions.
If the power has an odd numerator, we only take the positive solution.
Have students work an example, where they have to isolate the rational exponent term first.
After solving, have them summarize their steps.
Give students a simple equation in quadratic form like: x 4 5 x 2 4 0 . What do they
notice? Have them compare it with the quadratic equation x 2 5 x 4 0 . Since we know
how to solve the latter equation, can we somehow make a substitution to make the former
equation look like the latter?
It will most likely be difficult for the students to see this one on their own. Have students
think about it for a minute, then suggest (if no one else has) to let x 2 u . Tell students that
we want to write the equation in terms of u now.
What is x 4 then? Have students transform the equation to u 2 5u 4 0 .
Temporarily we should forget about the original equation and just focus on solving the u
equation. How do we solve this? What type of equation is it?
Have students solve for u. Then ask, are we finished? Did we solve the original equation?
NO! We needed x not u. How do we get x? What is the relationship between u and x? The
substitution linked the two variables.
Have students substitute each value of u to find x.
Give students another example to try, but give different power of x like x 6 and x3 .
Ask students if there is a pattern to what u is.
What is absolute value? Can more than one number have the same absolute value? Give an
example and explain your reasoning. What number is the exception?
For |x| = 5, what numbers could replace the x? How many solutions are there? Write
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equations for these solutions.
For |x + 1| = 5, what two numbers could x be? Using our equations for the example above,
write two equations to solve this problem.
Write an absolute value equation where you might have only one solution.
Write an absolute value equation where you might have no solution.
Now give the students an equation where the absolute value is not isolated. What needs to
be done first to solve |2x + 3|+ 4 = 9? Solve and check.
Now, if |x| =|2| what must be true about x? Must x = 2? Write 2 equations for x.
If x = 2 or x = -2, then how can we solve an equation like |x+2|=|2x – 3| using opposites?
Ask students to solve and check both answers.
Systems of Equations
Overview of Objectives, students should be able to:
1. Determine if a given ordered pair is a solution to a
system of linear equations.
2. Solve systems of linear equations using graphing.
3. Solve systems of linear equations by substitution.
4. Solve systems of linear equations by addition.
5. Identify systems that have no solution or infinitely
many solutions.
6. Solve systems of linear equations in three variables.
Objectives:
Determine if a given ordered pair is a solution to
a system of linear equations.
Solve systems of linear equations using graphing
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Main Overarching Questions:
1.
2.
3.
4.
5.
When is an ordered pair a solution of a system?
How do you solve systems of equations by graphing?
How do you solve systems of equations by algebraic methods?
Compare and contrast the methods of solving systems for efficiency and accuracy.
How do you determine when a system has no solution or infinitely many solutions?
Activities and Questions to ask students:
Present two linear equations and have students substitute an ordered pair into x and y.
What is meant by the term “solution to an equation”?
Does the ordered pair create true or false statements?
Is this point a solution for both of these equations?
When is an ordered a solution of a system of linear equations?
Students will graph 3 systems of linear equations: a pair of intersecting lines, a pair of
parallel lines, and two equations that are the same line.
Direct students to compare the three systems: describe the type of lines, describe how many
points they have in common, and compare the equations within each system
Students may present their results.
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Solve systems of linear equations by
substitution.
Solve systems of linear equations by addition
Select the most efficient method for solving a system
of linear equations.
Identify systems that have no solution or infinitely
many solutions.
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Have students summarize their results.
What does substitution mean? What can be substituted without changing the solution of an
equation?
If 2 equations are solved for y like y 2x 1 and y x 4 , can you say
that 2 x 1 x 4 ? Why or why not?
Have students solve for x.
Now that you can solve for x, how can you find y?
For students who struggle with substitution method, try the above method.
Demonstrate technique of solving one equation for a variable. How can we use substitution
to combine these two equations into a single equation with one variable?
How does this help us solve the system? How do you find the second variable?
What happens if you substitute into the wrong variable?
How do you add two equations? What parts can you add?
Ask students to add two given equations (where a variable will cancel). How does this help
us solve the system? How do you find the second variable?
Ask students to add two given equations where a variable does not cancel. Does this help us
solve the system? Why not? What needs to happen? What is it about the coefficients that
make a variable cancel when adding?
How can we change an equation so that a variable will cancel when we add? IF we change
one part of the equation, what must be done to the remaining terms in the equation?
Introduce or reemphasize term: equivalent equations
Have students compare/contrast the 3 methods of solving a system.
Do lines always intersect at integer points?
Can you always read the coordinates of the intersection on a graph?
Can you determine fraction solutions when solving using algebraic methods?
Ask students to make a conclusion about the efficiency and accuracy of each method.
Give students a system with no solution and ask them to use either the substitution or
addition method to solve.
What happens to the variables? What kind of statement is left? Is it true or false? Ask
students to solve each equation for y and compare the slopes and y-intercepts. What kind of
lines are they and how many solutions are there for this system?
Give students a system with infinitely many solutions and ask them to solve using addition or
substitution method.
What happens to the variables and the constants? What statement is left? True or false?
Ask students to solve both equations for y and to make a conclusion about the type of lines
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and the number of solutions.
Students should see this is an extension of solving systems of linear equations in two
variables.
Begin by giving students a system of linear equations in three variables and a numeric
ordered triple in the form ( x, y, z ) . How do the numbers and variables correspond? How
can we verify the ordered triple satisfies the system?
Ask students to think about how we solved systems in two variables. Is there a way we can
transform the three variable system to a two variable system? To facilitate the discussion it
would help to give a simple linear system in three variables, where one variable is easily
cleared.
Some students will suggest eliminating one variable using two equations. However, ask
them how many equations are needed to solve a system of two variables. How can we get
another equation in terms of the two remaining variables?
Have students conclude they need to eliminate one variable from two PAIRS of equations
resulting in two equations in terms of the two remaining variables.
Now that we have two equations in terms of two variables, how do we solve? If students
have trouble, have them discuss the ways we solved systems of equations earlier.
Students should now be able to solve and get numeric values for the two variables. How do
we get the value of the third variable we eliminated before? Ask students to specific about
which equation they use to find the third variable. Does it matter? Will the answer be
different?
Have students check their ordered triple in the original system.
Solve systems of linear equations in three variables.
Solving Linear and Absolute Value Inequalities
Overview of Objectives, students should be able to:
1. Use interval notation to represent solutions to
inequalities in one variable.
2. Find intersections and unions of intervals.
3. Solve linear inequalities in one variable.
4. Recognize linear inequalities that have no solution
or infinitely many solutions.
5. Solve compound inequalities.
6. Solve absolute value inequalities.
Objectives:
Use interval notation to represent solutions to
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Main Overarching Questions:
1. How do you solve and graph inequalities in one variable?
2. How do you determine the number of solutions to a linear inequality or if no solution exists?
3. How do you use set notation and interval notation to express the solutions of linear
inequalities?
Activities and Questions to ask students:
If x 3 , what number(s) does this x stand for? What is the least number included? Is 3
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inequalities in one variable.
Find intersections and unions of intervals.
Solve linear inequalities in one variable.
Recognize linear inequalities that have no
solution or infinitely many solutions.
Solve compound inequalities.
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included? Why or why not?
We use (3, ) to show all real numbers great than 3. How can we show all real numbers >
7? … > -2?
We use ( , 4) to show all real numbers less than 4. How do you use interval notation to
show all real numbers < 8? …< -3?
If x is greater than OR EQUAL TO 5, then using interval notation we write [5, ) and if x is less
than OR EQUAL TO 2, we write ( , 2] . How do you express all real numbers less than or
equal to 7 in interval notation?
Begin by giving two intervals that overlap over some interval. Describe as the intersection
or overlap of two sets. How can we tell where the two sets overlap? Is there a visual way we
can accomplish this?
Describe as the union or total collection of both intervals. Have students practice finding
unions of two intervals.
If necessary, review properties used in solving equations.
How would we solve x 2 6 ? How can we use these properties to solve x 2 6
How is the solution to an inequality different from a solution to an equation? How can we
check our solutions?
Ask students to solve 3x 9 and check their solution. If students fail to change < to >, ask
why their solution does not work when checked?
Or ask students to divide or multiply both sides of a true inequality like 2< 4 by -1. Is this
still a true inequality? Why or why not?
What must be done to make the solution of 3x 9 work? After what other operation will
you need to switch the inequality sign?
Give students more involved problems to work, including a linear equality that contains
fractions.
Can you think of a number for x so that x x 5 is a true statement? Try a positive, a
negative, and zero. Will x 5 always be greater than x? How many solutions will this
inequality have? If we did not see this was a special type of inequality, what would have
happened if we solved the inequality for x? 0 5 , which is always a FALSE statement.
Can you think of a number for x so that x x 5 is a true statement? Can you think of more
numbers? How many solutions will this inequality have? If we did not see this was a special
type of inequality, what would have happened if we solved the inequality for x? 0 5 ,
which is always a TRUE statement.
Recall that “and” statements and “between statements” are the same. If 5 x 8 , what
numbers could x represent? How could we write this is as two inequalities?
Now that students understand this type of inequality, we need to solve them. Give a simple
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Solve absolute value inequalities.
o Solve inequalities of the type Ax
B
C
B
C
o
Solve inequalities of the type Ax
o
Recognize absolute value inequalities with
no solution or all real numbers as
solutions.
example with a step performed and ask students what happened. For example,
8 2 x 10 becomes 4 x 5 .
Repeat with addition and subtraction. Ask students what rule(s) do we need to follow to
solve this type of inequality. What are we trying to isolate?
Direct students to draw a number line and label and graph numbers whose absolute value is
less than 5. If students graph only integers or only positive numbers, ask them if -2.5 has an
absolute value < 5.
Ask students to draw a conclusion about the numbers whose absolute value is less than 5.
Ask students to write an “and” or “between” statement to describe their conclusion for all
numbers, x, such that |x| < 5.
Using this interpretation, how can we solve |x +2|< 5?
Direct students to draw a number line and label and graph numbers whose absolute value is
greater than 3. |x| > 3
Ask students to describe the solutions using inequalities and then write as a compound
inequality. Discuss why the word “or” is used instead of “and”.
How is the solution to a “>” problem different from a “<” problem?
Can you use a “between” statement for this problem? Why or why not?
Using a compound inequality, how can we solve |x + 2| > 3?
What happens if we solve | x | 2 ? What values of x make this a true statement? What is
the solution set?
What happens if we solve | x | 2 ? What values of x make this a true statement? What is
the solution set?
Ask students if they see a pattern. Have them write a generalization to spot these special
cases.
Graphing Inequalities
Overview of Objectives, students should be able to:
1. Graph (solve) a linear inequality in two variables.
2. Graph (solve) a system of linear equalities in two
variables.
Objectives:
Graph (solve) a linear inequality in two variables.
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Main Overarching Questions:
1. How do you solve and graph a linear inequality in two variables?
2. What is a boundary line?
3. How do you graph a system of linear inequalities in two variables?
Activities and Questions to ask students:
Background knowledge: How do you graph y 2x 1? Graph the equation.
Ask students to name a point that would make y 2x 1. Shade the side of the line where
the point lies.
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Graph (solve) a system of linear equalities in two
variables.
Graph the line again on a new grid and ask students to name a point that makes y 2x 1 .
Shade the side where the point lies.
Define boundary line and the difference between dashed and solid boundary lines.
What is the difference in the solution to y 2x 1 and y 2x 1 ?
Background knowledge. Review graphing a system of linear equations. What is the solution
to a system with intersecting lines?
Ask students to graph two inequalities that intersect on the same grid. What points do the
two graphs have in common? How many? How does the graph show this?
Ask student to graph two inequalities on the same grid that do not overlap. What points do
the inequalities have in common? What is the solution to this system?
Problem Solving and Modeling
Overview of Objectives, students should be able to:
1.
2.
Use linear equations to solve problems
Solve a formula for a variable.
Objectives:
Use linear equations to solve problems
Solve a formula for a variable.
Main Overarching Questions:
1. How do you setup and solve a linear equation to solve a problem?
2. How do you solve for a variable in a formula?
Activities and Questions to ask students:
In these types of sections, it’s best to begin with a word problem and discuss solutions with
the students. For example consider this problem: A new car is worth $24000 but
depreciates by $3000 per year.
First we want to determine a model for the worth of the car after x number of years. How
much is the car worth in year 0? What about at the end of the first year? 2nd year? What’s
the pattern? Can you write an equation to describe the worth of the car after x years?
Discuss general problem solving strategies with students.
2
Give an example of a formula like E mc . What is different about this equation? How
many variables are there?
Next ask the students to solve the formula for m. What do you need to move? How do you
move it? Can you get a numeric value for m? Why not?
Have students practice solving for a variable in a given formula.
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