Download Pre-Algebra Notes – Unit Three: Multi-Step Equations and

Pre-Algebra Notes – Unit Three:
Multi-Step Equations and Inequalities (optional)
CCSD Teachers note: CCSD syllabus objectives (2.8)The student will solve multi-step
inequalities and (2.9)The student will graph solutions to multi-step inequalities have been
eliminated for 2011-2012. However, we have chosen to include the notes as some teachers have
expressed that they will still include the concepts in their classrooms.
A note to substitute teachers: pre-algebra teachers agree that all units of study are important, but
understanding this unit seems to be a key as to whether or not a student is successful with the
following chapters. While it is important to maintain the pace of the CCSD benchmark calendar,
veteran teachers warn us not to rush through this material.
Solving Two-Step Equations
Syllabus Objective: (2.4) The student will solve two-step equations.
The general strategy for solving a multi-step equation in one variable is to rewrite the equation in
ax + b =
c format (where a, b, and c are real numbers), then solve the equation by isolating the
variable using the Order of Operations in reverse and using the opposite operation. (Remember
the analogy to unwrapping a gift…)
1.
2.
3.
4.
Order of Operations
Parentheses (Grouping)
Exponents
Multiply/Divide, left to right
Add/Subtract, left to right
Evaluating an arithmetic expression using the Order of Operations will suggest how we might go
about solving equations in the ax + b =
c format.
To evaluate an arithmetic expression such as 4 + 2 ⋅ 5 , we’d use the Order of Operations.
=
4 + 2⋅5
=
4 + 10
14
First we multiply, 2 ⋅ 5
Second we add, 4 + 10
Now, rewriting that expression, we have 2 ⋅ 5 + 4 =
14 , a form that leads to equations written in
the form ax + b = c. If I replace 5 with n, I have
2⋅n + 4 =
14 or
2n + 4 =
14,
an equation in the ax + b =
c format.
McDougal Littell, Chapter 3
HS Pre-Algebra , Unit 03: Multi-Step Equations and Inequalities
Revised 2013 - CCSS
Page 1 of 14
To solve that equation, I am going to “undo” the expression “2n + 4”. I will isolate the variable
by using the Order of Operations in reverse and using the opposite operation.
That translates to getting rid of any addition or subtraction first, then getting rid of any
multiplication or division next. Undoing the expression and isolating the variable results in
finding the value of n.
This is what it looks like:
2n + 4 14
2n + 4 14
=
2n + 4 − 4 =
14 − 4
subtract 4 from each side to "undo" the addition
− 4 =−4
2n 10
2n 10
=
2n 10
2n 10
divide by 2 to "undo" the mulitplication
=
=
2
2
2
2
n 5=
n 5
Check your solution by substituting the answer back into the original equation.
2n + 4 =
14
2(5) + 4 =
14
10 + 4 =
14
14 = 14
Example:
original equation
substitute '5' for 'n'
true statement, so my solution is correct
Solve for x, 3x − 4 =
17 .
Using the general strategy, we always want to “undo” whatever has been done in reverse
order. We will undo the subtracting first by adding, and then undo the multiplication by
dividing.
=
3x − 4 17
=
or
3x − 4 17
3x − 4 + 4 =
17 + 4
+ 4 =+4
=
3x
3x
=
3
=
x
Check:
21
=
3x 21
21
3x 21
=
3
3
3
7=
x 7
3x − 4 =
17
3(7) − 4 =
17
21 − 4 =
17
17 = 17
McDougal Littell, Chapter 3
HS Pre-Algebra , Unit 03: Multi-Step Equations and Inequalities
Revised 2013 - CCSS
Page 2 of 14
Example:
Solve for x,
x
+ 5 = 12
4
x
x
=
+ 5 12
=
or
+ 5 12
4
4
x
+ 5 − 5 =12 − 5
− 5 =−5
4
x
x
= 7= 7
4
4
x
x
(4)   = (4)(7)
(4)   = (4)(7)
4
4
x 28
x 28
=
=
Check:
x
+5=
12
4
(28)
+5=
12
4
7+5=
12
12 = 12
***NOTE: Knowing how to solve equations in the ax + b =
c format is extremely important for
success in algebra. All other equations will be solved by converting equations to ax + b =
c . To
solve systems of equations, we rewrite the equations into one equation in the ax + b =
c form and
solve. (In the student’s algebraic future, to solve quadratic equations we will rewrite the
equation into factors using ax + b =
c , then solve the resulting equation letting c = 0 . It is
important that students are comfortable solving equations in the ax + b =
c format.)
Solving Equations that Are NOT in the ax + b =
c Format
Syllabus Objective: (2.5) The student will solve equations with like terms and parentheses.
The general strategy for solving equations NOT in the ax + b =
c format is to rewrite the
equation in ax + b =
c format using the Properties of Real Numbers.
An important problem solving/learning strategy is to take a problem that you don’t know how to
do and transform that into a problem that you know how to solve. Right now, we know how to
solve problems such as ax + b =
c . I cannot make that problem more difficult, but what I can do
is make it look different.
For instance, if I asked you to solve 5(2 x + 3) − 4 =
21 , that problem looks different from the
2x + 4 =
14 that we just solved by using the Order of Operations in reverse. The question you
have to ask is “what is physically different in these two problems?”
McDougal Littell, Chapter 3
HS Pre-Algebra , Unit 03: Multi-Step Equations and Inequalities
Revised 2013 - CCSS
Page 3 of 14
The answer is the new problem has parentheses. That means to make these problems look alike,
we need to get rid of those parentheses. We’ll do that by using the distributive property.
5(2 x + 3) − 4 =
21
10 x + 15 − 4 =
21
10 x + 11 =
21
we applied the distributive property
we combined like terms
Now I have converted 5(2 x + 3) − 4 =
c format as 10 x + 11 =
21 . We recognize
21 into ax + b =
this format and know how to solve it.
=
=
or
10 x + 11
21
10 x + 11 21
− 11 =
−11
10 x + 11 − 11 =
21 − 11
=
=
10 x 10
10 x 10
10 x 10
10 x 10
= =
10 10
10 10
=
x 1=
x 1
Check:
5(2 x + 3) − 4 =
21
5 [ 2(1) + 3] − 4 =
21
5 [ 2 + 3] − 4 =
21
5 [5] − 4 =
21
25 − 4 =
21
21 = 21
Example:
Solve for x, 4(3x − 2) + 5 =
45 .
The parentheses make this problem look different than the ax + b = c problems.
Get rid of the parentheses by using the distributive property, and then combine terms.
4(3x − 2) + 5 =
45
12 x − 8 + 5 =
45
12 x − 3 =
45
12 x − 3 + 3 = 45 + 3
12 x = 48
12 x 48
=
12 12
x=4
We will leave the check to you!
McDougal Littell, Chapter 3
HS Pre-Algebra , Unit 03: Multi-Step Equations and Inequalities
Revised 2013 - CCSS
Page 4 of 14
Caution! Note that when we solved these equations we got rid of the parentheses first.
Students will often want to get rid of the addition/subtraction first because they are using
the Order of Operations in reverse. The reason we get rid of the parentheses first is because the
strategy is to rewrite the equation in ax + b =
c format before using the Order of Operations in
reverse.
Reading an Inequality (optional; see note on page 1)
We know “>” means “greater than” when we read from left to right. So 5 > 4 is read “five is
greater than 4”. When I read from right to left, the way the inequality is read changes, and it
also means “four is less than five”. This could be written 4 < 5.
In the problems that follow, we may have a solution like −4 > x . Reading from left to right, we
say “negative four is greater than x”. Generally in algebra, the convention is to always read the
variable first. So, rather than saying “negative four is greater than x”, we would say “x is less
than negative four” and rewrite it as “ x < −4 .
That suggests these statements are equivalent:
−4 > x
⇔
x < −4
Example: Which graph represents the solution set of the inequality −2 > x ? (Note that
a problem similar to this appeared in the CRT Instructional Materials.)
A
B
C
D
−5
0
5
−5
0
5
−5
0
5
−5
0
5
It is a common error for students to now select “C” as the correct answer.
You may want to encourage students to rewrite the inequality with the
variable on the left side, as x < −2 , which more clearly shows the answer of
“A”.
McDougal Littell, Chapter 3
HS Pre-Algebra , Unit 03: Multi-Step Equations and Inequalities
Revised 2013 - CCSS
Page 5 of 14
Solving Linear Equations with Variables on Both Sides
Syllabus Objective: (2.6) The student will solve equations with variables on both sides.
Using the same strategy, we rewrite the equation so it is in ax + b =
c format.
Example:
Solve for x, 5 x + 2 = 3x + 14 .
We ask ourselves, “what is physically different in this problem?” This equation is
different because there are variables on both sides of the equation. The strategy is
to rewrite the equation in ax + b =
c format which calls for variables on only one
side of the equation.
What’s different? There is a 3x on the other side of the equation. How do we get
rid of the addition of 3x?
5x + 2 = 3x + 14
5x + 2 – 3x = 3x + 14 – 3x
5x – 3x + 2 = 3x – 3x + 14
2x + 2 = 14
Subtraction Property of Equality
Commutative Property of Addition
Combining like terms
Now the equation is in ax + b =
c format.
2x + 2 = 14
2x + 2 – 2 = 14 – 2
Subtraction Property of Equality
2x = 12
Arithmetic fact
2x 12
Division Property of Equality
=
2
2
x=6
Arithmetic fact
Let’s try a few more examples:
5 x − 17 = 3x − 9
−3x
=
−3 x
2 x − 17 =
−9
+17 =
+17
2x = 8
2x 8
=
2 2
x=4
−2 x − 17 =6 − x
+x
=
+x
−1x − 17 = 6
+17 =
+17
−1 x =
23
−1x 23
=
−1 −1
x = −23
3m + 8 − 5m = 9 + 4m + 29
−2m + 8 = 4m + 38
−2m − 4m + 8 = 4m − 4m + 38
−6 m + 8 =
38
−8 =−8
−6 m =
30
−6m 30
=
−6
−6
m = −5
Now, I can make equations longer, but I cannot make them more difficult!
McDougal Littell, Chapter 3
HS Pre-Algebra , Unit 03: Multi-Step Equations and Inequalities
Revised 2013 - CCSS
Page 6 of 14
CCSS 8.EE.7b: Solve linear equations with rational number coefficients, including equations
whose solutions require expanding expressions using the distributive property and collecting
like terms.
Example:
Solve for x, 5( x + 2) − 3= 3( x − 1) − 2 x .
In this problem, there are parentheses and variables on both sides of the equation.
It is clearly a longer problem. The strategy remains the same: rewrite the
equation in ax + b =
c format, and then isolate the variable by using the Order of
Operations in reverse using the opposite operation.
Physically, this equation looks different because there are parentheses, so let’s get
rid of them by using the distributive property.
5( x + 2) − 3= 3( x − 1) − 2 x
5 x + 10 − 3 = 3x − 3 − 2 x
5x + 7 = x − 3
5x + 7 − x = x − 3 − x
4 x + 7 =−3
4 x + 7 − 7 =−3 − 7
4 x = −10
4 x −10
=
4
4
−5
=
x
or − 2.5
2
Distributive Property
Combine like terms
Subtraction Property of Equality
Combine like terms
Addition Property of Equality
Simplify
Division Property of Equality
Simplify
Caution! When solving equations, we often write arithmetic expressions such as
−3 + ( −7) as −3 − 7 as we did in the above problem. Students need to be reminded that
when there is not an “extra” minus sign, the computation is understood to be an addition
problem.
Examples:
( +5) + ( +6) = 5 + 6 = 11
( −5) + ( −6) =−5 − 6 =−11
( −8) + ( +3) =−8 + 3 =−5
Emphasizing this will help students to solve equations correctly. If not taught, students
often solve the equation correctly only to make an arithmetic mistake—then they think
the problem they are encountering is algebraic in nature (instead of arithmetic).
Also make sure students understand simplifying expressions with several negatives.
Examples:
5 − ( −4) = 5 + 4 = 9
−3 − ( −8) =−3 + 8 =5
−2 − 5 =−2 + ( −5) =−7
McDougal Littell, Chapter 3
HS Pre-Algebra , Unit 03: Multi-Step Equations and Inequalities
Revised 2013 - CCSS
Page 7 of 14
Solving Equations with “No Solutions” or “Infinitely Many” as a Solution
CCSS 8.EE.7a: Give examples of linear equations in one variable with one solution, infinitely
many solutions, or no solutions. Show whether a linear equation in one variable has one
solution, infinitely many solutions or no solutions by successively transforming the given
equation into simpler forms, until an equivalent equations of the form=
x a=
, a a, or
=
a b
results (where a and b are different numbers).
When you solve an equation, you may find something unusual happens: when you subtract the
variable from both sides (in an effort to get the variable term on one side of the equation), no
variable term remains! Your solution is either “no solution” or “infinitely many”. Look at the
following examples.
Example:
Solve 3 ( 3x − 1) =
9x .
3 ( 3x − 1) =
9x
9x − 3 =
9x
9 x − 3 =9 x
−9 x
=
−9 x
−3 =0
Notice we could stop now if we recognized that it is
impossible for a number 9x to be equal to 3 less than
itself. If we continue…..
….we still have a statement that is not true.
So the equation is said to have NO SOLUTION.
Example:
Solve 8 x −=
2 2 ( 4 x − 1) .
8 x −=
2 2 ( 4 x − 1)
8x − 2 = 8x − 2
8x − 2 = 8x − 2
Notice that this statement would be true no matter what
value we substitute for x. We could go further….
−8 x =
−8 x
−2 =−2
McDougal Littell, Chapter 3
…. We still get a true statement. So the equation is said
to have a solution of INFINITELY MANY.
HS Pre-Algebra , Unit 03: Multi-Step Equations and Inequalities
Revised 2013 - CCSS
Page 8 of 14
Solving and Graphing Inequalities (optional; refer to note on page 1)
Syllabus Objectives: (1.4) The student will solve inequalities involving integers. (2.7) The
student will graph the solution set of a linear equation in one variable. (2.8) The student will
solve multi-step inequalities. (2.9) The student will graph solutions to multi-step inequalities.
We can solve linear inequalities the same way we solve linear equations. We write the
inequalities in the ax + b > c format, then we use the Order of Operations in reverse using the
opposite operation. Linear inequalities look like linear equations with the exception they have a
greater than (>), greater than or equal to ( ≥ ), less than (<) or less than or equal to ( ≤ ) sign rather
than just an equal sign.
Examples:
Linear Equation
3x – 2 = 10
Linear Inequality
3x – 2 > 10
Let’s solve each and graph the solution. Solving them both, we have:
3x − 2 10
=
3x − 2 + 2 = 10 + 2
3x 12
=
3x 12
=
3
3
=
x 4
3x − 2 > 10
3x − 2 + 2 > 10 + 2
3x > 12
3x 12
>
3
3
x>4
To graph x = 4, we place a point on the number line at 4.
−5
0
5
To graph x > 4, we will place an open circle at 4 because 4 is not included in the solution. We
will “shade” or draw a solid line with an arrow to the right, which indicates that all the numbers
greater than 4 are part of the solution set.
−5
0
5
If the inequality contained an equal sign, 3x – 2 ≥ 10, then everything would be done the same in
terms of solving the inequality, except the answer and graph would look a little different because
it would include 4 as part of the solution set. To show 4 was included, we would shade it.
The solution is x ≥ 4.
−5
0
5
It may appear that the Properties of Equalities that we have studied will also work for Properties
of Inequalities. They do for addition and subtraction, but for multiplication and division, they
hold true for positive numbers only.
McDougal Littell, Chapter 3
HS Pre-Algebra , Unit 03: Multi-Step Equations and Inequalities
Revised 2013 - CCSS
Page 9 of 14
Example:
Find and graph the solution set for 3x − 12 ≤ 13.
3x − 2 ≤ 13
3x − 2 + 2 ≤ 13 + 2
3x ≤ 15
3x 15
≤
3
3
x≤5
−5
Addition Property of Inequality
Combine like terms
Division Property of Inequality
Simplify
0
5
Again, I can’t make these problems more difficult, I can only make them longer as I did with
linear equations.
But we need to consider what happens to the value of an inequality if we multiply or divide by a
negative number. What would happen if I had an inequality such as 6 > −5 and multiplied both
sides of the inequality by –2?
6 > −5
( −2)6 > ( −2)( −5)
− 12 > 10
This is a true statement.
Is this true?
This is NOT true.
If we look at the number line, we notice that –12 would be to the left of 10 on the number line.
Numbers to the left are smaller, so −12 < 10 . It appears when we multiply an inequality by a
negative number, to make the statement true, the order of the inequality must be reversed.
Note the inequality below. When both sides are divided by a negative number, the order of the
inequality must also be reversed.
10 > −15
10 −15
>
−5 −5
−2 > 3
This is true.
Is this true?
This is NOT true!
What is true: −2 < 3 ; we had to reverse the inequality to make a true statement.
Multiplication Property of Inequality
If a < b and c > 0, then ac < bc.
If a < b and c < 0, then ac > bc.
In other words, when you multiply an inequality by a negative number (c < 0), the
direction of the inequality changes.
McDougal Littell, Chapter 3
HS Pre-Algebra , Unit 03: Multi-Step Equations and Inequalities
Revised 2013 - CCSS
Page 10 of 14
Division Property of Inequality
a b
< .
c c
a b
If a < b and c < 0, then > .
c c
If a < b and c > 0, then
That is, when you divide an inequality by a negative number, the direction of the
inequality changes.
Example:
Find the solution set for −3x − 2 > 10 .
− 3x − 2 > 10
−3x − 2 + 2 > 10 + 2
− 3x > 12
−3x 12
<
−3 −3
x < −4
Notice that when we divided by a negative number, we reversed the order
of the inequality.
We could have done the same problem without multiplying or dividing by a negative number by
keeping the variable positive. That would result in not having to reverse the inequality. Let’s
look at the same problem.
Example:
Find the solution set for −3x − 2 > 10 .
Rather than having the variables on the left side and have the coefficient negative, I could
put the variables on the right side (add 3x to both sides) that would result in a positive
coefficient.
− 3x − 2 > 10
3x − 3x − 2 > 3x + 10
− 2 > 3x + 10
− 2 − 10 > 3x + 10 − 10
− 12 > 3x
−12 3x
>
3
3
or
−4> x
x < −4 *
McDougal Littell, Chapter 3
HS Pre-Algebra , Unit 03: Multi-Step Equations and Inequalities
Revised 2013 - CCSS
Page 11 of 14
In both cases, we have the same answer. (*An explanation is in a preceding section Reading an
Inequality.) In the second example, since I kept my variables positive, I did not have to reverse
the order of the inequality.
Bottom line, you solve linear inequalities the same way you solve linear equations: write them
in ax + b =
c format, then use the Order of Operations in reverse using the opposite operation.
Remember, if you don’t keep the coefficients of the variable positive, then you will likely have
to multiply or divide by a negative number that will require you to change the order of the
inequality.
Solving and Graphing Equations and Inequalities with Absolute Value
CCSD Teachers note: This is not a CCSD syllabus objective, and the State Standard:
(2.8.2)The student will solve and graphically represent equations and inequalities in one
variable, including absolute value has been partially eliminated for 2011-2012. The Nevada
State Grade 8 Transition Guide states “Continue to solve equations and inequalities in one
variable. Graphing inequalities is in the CCSS in Grade 7 and High School.” The
following material has been included for your general information.
Some equations will involve absolute value. You must remember that two numbers can have the
same absolute value.
Example: Find and graph the solution set for x = 3. (A similar question was included
on the CRT Instructional Materials.)
Using the definition of absolute value, we think “what are the points that are
each a distance of 3 from zero?” We know that there are 2 values for x that
make this a true statement: x = 3 or x = −3. The graph would be:
−5
0
5
Example: Find and graph the solutions set for x − 2 =
3.
First, we must isolate the absolute value.
3
x −2=
+2 =+2
x =5
McDougal Littell, Chapter 3
HS Pre-Algebra , Unit 03: Multi-Step Equations and Inequalities
Revised 2013 - CCSS
Page 12 of 14
We know that for x = 5 to be true, x = 5 or x = −5 because
=
5 5 and =
−5 5 .
−5
5
0
Let’s check our solution by substituting the answers back into the original
equation.
x −2=
3 is the original equation
=
−5 − 2 3
=
substitute answers
5 −2 3
=
5−2 3
=
5−2 3
3 = 3
3 = 3
Always remember to isolate the absolute value term before breaking the problem into two
equations.
5.
Example: Solve 2 x − 1 =
First, let’s isolate the 2x .
2 x − 1 =5
+1 =+1
2x = 6
If 2 x = 6, then either
2x =
−6
or
2x =
6
2 x −6
2x 6
= =
2
2
2 2
x=
−3
or
x=
3
−5
5
0
Let’s consider graphing simple absolute value inequalities. If we were asked to graph the
solution to x < 2 , we can think “what are the values of x that are less than 2 units away from
zero?” It would be the values between −2 and 2.
−5
McDougal Littell, Chapter 3
0
5
HS Pre-Algebra , Unit 03: Multi-Step Equations and Inequalities
Revised 2013 - CCSS
Page 13 of 14
Likewise, let’s reason what the graph of x > 2 would look like. It would include the values of x
that are more than 2 units away from zero. That would be values greater than 2 or less than −2 .
−5
0
5
0
5
0
5
Example: Graph the solution for x ≤ 3 .
−5
Example: Graph the solution for x ≥ 3 .
−5
McDougal Littell, Chapter 3
HS Pre-Algebra , Unit 03: Multi-Step Equations and Inequalities
Revised 2013 - CCSS
Page 14 of 14