Download Solving Equations and Inequalities (One Step)

Solving 1-step equations and inequalities
SM1 Section 1-4
Date______________
Objective: Students will solve single-step equations and inequalities.
An Equation states that two expressions are equal. An Inequality shows that two
expressions are not equal. Some equations and inequalities, called Identities, are true for all
values of the variables they contain. For example, you can use any value for x in the following
and they remain true:
3 + 22 = 7
x < x +1
2(x + 5) = 2x + 10
Other equations and inequalities are called Conditional if they are not true for all values
of the variables they contain. There is only one value that can be used to make the statement
true. Can you find them?
x+2=5
3£ a-5
Finding the value of the variable which makes the statement true is called solving the
equation (or inequality), and these values of the variables are solutions. What would be the
solutions of the two expressions above?
To make sure a number is a solution to an algebraic expression you can check by substituting it
into the original equation.
Example:
Which of the following numbers are solutions to the equation:
a. x = 2
3x - 5 = 4
b. x = 3
When we began our work with equations, we found that there was a single answer that
would make our statements true. For example, if we are given the equation x - 4 = 5 , we know
there exists only one number that we can substitute for x to get 5. We learned to use the
properties of algebra to isolate the variable. In this case, we would add four to both sides
because that is the inverse operation to subtracting 4. When we add four to both sides we find
the value that makes this original statement true, x = 9 . We can check this answer by
substituting nine for x in the original equation and we see that 9 – 4 = 5 is a true statement so 9 is
the only answer.
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The statement, “nine is greater than negative two” is an example of an inequality.
Inequalities are used in instances when more than one number can give a true statement. For
example, if we change the first inequality from 9 > – 2 to “a number greater than negative two”
we find there are many numbers that are greater than – 2. You have dealt with inequalities
throughout your school years. Many teachers have set a range for each grade in the grading
scale. The idea that 90% or better is an “A.” can be expressed as an inequality. Each student
scoring above 90 will expect to get an A in class. But the wording used also implies that a score
of 90 would also get an A. To make sure we include this value we use the inequality A ³ 90. In
this example, we have a symbol that combines the “equal” sign and the “greater than” sign. The
90 is the boundary. Similarly, £ would imply any number equal to or below the given boundary.
For example, F £ 70 would be true for any number less than or equal to 70 . The inequality
symbols are shown below.
>
<
³
£
¹
Greater than
Less than
Greater than or equal to
Less than or equal to
Is not equal to, or Does not equal
When we begin graphing simple equations such as x = 3, we placed a closed circle on the
number line at positive 3 as seen below.
If we are graphing a simple inequality such as x > 3, we must remember that a closed
circle at 3 indicates the inclusion of 3 in the answer, but our inequality wants every number that
is bigger than 3. We show this relationship with an open circle at our boundary (3) and then use
an arrow to show all of the answers that make the statement true. Since every number bigger
than 3 is part of the solution, we must find some way of showing all numbers without having to
list every one. We do this with an arrow as seen below.
If the inequality includes the boundary, (uses £ or ³ ) then we will have a closed circle at
the boundary and the appropriate arrow for all other numbers in the solution.
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Practice: Graph the following inequalities on the number line provided.
1. x < 5
2. x ³ -3
3. x £ 2
4. x > 6
Looking at how we represent all numbers that make the statement true, how do you think
we would graph the inequality x ¹ 4 ? Explain your reasoning:
In our last section we explored some of the ways that numbers behave. We introduced
the concepts of inverse operations, identity, and the equality properties of an equation. These
can be summarized as follows:
Identity Properties
Property
Identity
Inverse
a+0= a
...of addition
a + ( -a ) = 0,
-a + a = 0
…of multiplication
a ×1 = a
a b
× =1
b a
Explain these properties in your own words:
Properties of Equality
For Addition
For Subtraction
For Multiplication
For Division
if a = b, then a + c = b + c
if x + 5 = -2, then x + 5 - 5 = -2 - 5
b
b
if = -2, then 5 × = 5 × -2
5
5
d rt
if d = rt, then =
r r
Summarize the properties of equality in your own words:
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a×
1
=1
a
Solving an inequality is similar to solving an equation. We will use the inverse property
in order to obtain the identity elements for addition and multiplication. However, since
inequalities are showing two expressions that are not equal, the properties for equality do not
apply. There are times that we need to switch the direction of the inequality. The following
activity lets us explore when we are supposed to make this switch. This activity can be done
anytime you are not sure of the rules.
Activity: When do we change the sign of an inequality?
Part 1: Think of a number, any number, and write it below. Now think of a different number
and write it to the right of the first number. Now choose the appropriate inequality sign that
compares these two numbers. < (less than) > (greater than).
Now add 5 to both numbers and write the two sums under the original numbers.
For example, if I had chosen 2 and 18 I would have 2 < 18
+5 +5
7 23
Now write the appropriate inequality sign between these two sums. (in my example,
7 < 23) My inequality sign did not change direction. It is the same as the original inequality
sign. Check with your neighbors in class and see if anyone changed their inequality sign.
Rule for adding to inequalities (Circle the correct answer)
Do or Do Not change the inequality sign.
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Part 2: Think of a number, any number, and write it below. Now think of a different number
and write it to the right of the first number. Now choose the appropriate inequality sign that
compares these two
numbers. < (less than) > (greater than).
Now subtract 10 from both numbers and write the two differences under the original numbers.
For example, if I had chosen 2 and 18 I would have 2 < 18
– 10 – 10
–8 < 8
Now write the appropriate inequality sign between these two differences. (in my example
– 8 < 8) My inequality sign did not change direction. It is the same as the original inequality
sign. Check with your neighbors in class and see if anyone changed their inequality sign.
Rule for subtracting from inequalities (Circle the correct answer)
Do or Do Not change the inequality sign.
Part 3: Think of a number, any number, and write it below. Now think of a different number
and write it to the right of the first number. Now choose the appropriate inequality sign that
compares these two numbers. < (less than) > (greater than).
Now multiply both numbers by 10 and write the two products under the original numbers.
For example, if I had chosen 2 and 18 I would have 2 < 18
´10 ´10
20 < 180
Now write the appropriate inequality sign between these two products. (in my example,
20 < 180) My inequality sign did not change direction. It is the same as the original inequality
sign. Check with your neighbors in class and see if anyone changed their inequality sign.
Rule for multiplying a positive to inequalities (Circle the correct answer)
Do or Do Not change the inequality sign.
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Part 4: Think of a number, any number, and write it below. Now think of a different number
and write it to the right of the first number. Now choose the appropriate inequality sign that
compares these two numbers. < (less than) > (greater than).
Now multiply both numbers by – 2 and write the two products under the original numbers.
For example, if I had chosen 2 and 18 I would have 2 < 18
´ - 2 ´ -2
– 1 > – 36
Now write the appropriate inequality sign between these two products. (in my example –
1 > – 9) My inequality sign changed direction. It is opposite of the original inequality sign.
Check with your neighbors in class and see if anyone changed their inequality sign.
Rule for multiplying a negative to inequalities (Circle the correct answer)
Do or Do Not change the inequality sign.
In this example, be careful with the positive and negative signs. Remember that a
positive times a positive is positive, a positive times a negative is negative, a negative times a
positive is negative and a negative times a negative is positive.
Part 5: Think of a number, any number, and write it below. Now think of a different number
and write it to the right of the first number. Now choose the appropriate inequality sign that
compares these two numbers. < (less than) > (greater than).
Now divide both numbers by 2 and write the two quotients under the original numbers.
For example, if I had chosen 12 and 8 I would have 12 > 8
¸2 ¸ 2
6 > 4
Now write the appropriate inequality sign between these two products. (in my example,
6 > 4) My inequality sign did not change direction. It is the same as the original inequality sign.
Check with your neighbors in class and see if anyone changed their inequality sign.
Rule for dividing a positive to inequalities (Circle the correct answer)
Do or Do Not change the inequality sign.
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Part 6: Think of a number, any number, and write it below. Now think of a different number
and write it to the right of the first number. Now choose the appropriate inequality sign that
compares these two numbers. < (less than) > (greater than).
Now divide both numbers by – 2 and write the two products under the original numbers.
For example, if I had chosen 22 and 14 I would have 22 > 14
¸ - 2 ¸ -2
– 11 < – 7
Now write the appropriate inequality sign between these two products. (in my example, –
11 < – 7 ) My inequality sign changed direction. It is opposite of the original inequality sign.
Check with your neighbors in class and see if anyone changed their inequality sign.
Rule for dividing a negative to inequalities (Circle the correct answer)
Do or Do Not change the inequality sign.
Now that we know when to switch the direction of the inequality, we can work on solving
inequalities involving a single step.
Solve the inequality.
Example 1.
In this example, the single operation is adding 7. The inverse of adding 7 is to subtract 7.
We must do this to both sides in order to keep the value of the original inequality. Subtracting 7
from 7 gives the identity element of addition (0).
Notice that subtracting 7 does not require us to change the direction of the inequality so
the answer is “x is greater than 14”.
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Example 2.
In this example, the single operation is multiplying by 3. The inverse of multiplying by 3
is to divide by 3. We must do this to both sides in order to keep the value of the original
inequality. Dividing 3 by 3 gives the identity element of multiplication (1).
Dividing by a positive 3 does not require a change in direction of the inequality so the
answer is “x is less than 7”.
Example 3.
In this example, the single operation is multiplying by – 5. The inverse of multiplying by
– 5 is to divide by – 5. We must do this to both sides in order to keep the value of the original
inequality. Dividing – 5 by – 5 gives the identity element of multiplication (1).
Since we divided by a negative number, we know that we must change the direction of
the inequality so the answer is “x is greater than or equal to negative 7”.
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