2.8 Solving Linear
Inequalities
Definition.
An inequality is an algebraic expression related by
< “is less than,” ≤ “is less than or equal to,”
> “is greater than,” or
≥ “is greater than or equal to.”
Linear Inequality in One Variable
A linear inequality in one variable can be written in the form
Ax  B  C , Ax  B  C , Ax  B  C , or Ax  B  C ,
where A, B, and C represent real numbers, and A ≠ 0.
We solve an inequality by finding all real number solutions of it. For example,
the solution set {x | x ≤ 2} includes all real numbers that are less than or
equal to 2, not just the integers less than or equal to 2.
Slide 2.8-3
Objective 1
Graph intervals on a number line.
Slide 2.8-4
Graph intervals on a number line.
Graphing is a good way to show the solution set of an inequality.
We graph all the real numbers belonging to the set {x | x ≤ 2} by placing a
square bracket at 2 on a number line and drawing an arrow extending from
the bracket to the left (to represent the fact that all numbers less than 2 are
also part of the graph).
Slide 2.8-5
Graph intervals on a number line. (cont’d)
The set of numbers less than or equal to 2 is an example of an interval on
the number line. To write intervals, we use interval notation. For
example, the interval of all numbers less than or equal to 2 is written
(−∞, 2].
The negative infinity symbol −∞ does not indicate a number, but shows that
the interval includes all real numbers less than 2.
As on the number line, the square bracket indicates that 2 is part of the
solution.
A parentheses is always used next to the infinity symbol. The set of real
numbers is written as (−∞, ∞).
Slide 2.8-6
CLASSROOM
EXAMPLE 1
Graphing Intervals on a Number Line
Write each inequality in interval notation, and graph the interval.
x3
Solution:
[3,  )
4  x
Solution:
(, 4)
Slide 2.8-7
Graph intervals on a number line. (cont’d)
Keep the following important concepts regarding interval notation in mind:
1. A parenthesis indicates that an endpoint is not included in a solution set.
2. A bracket indicates that an endpoint is included in a solution set.
3. A parenthesis is always used next to an infinity symbol, −∞ or ∞.
4. The set of all real numbers is written in interval notation as (−∞,∞).
Some texts use a solid circle ● rather than a square bracket to indicate the
endpoint is included in a number line graph. An open circle is used to
indicate noninclusion, rather than a parentheses.
Slide 2.8-8
Objective 2
Use the addition property of inequality.
Slide 2.8-9
Use the addition property of inequality.
Addition Property of Inequality
If A, B, and C represent real numbers, then the inequalities
A B
and
AC  B C
Have exactly the same solutions.
That is, the same number may be added to each side of an inequality
without changing the solutions.
As with the addition property of equality, the same number may be
subtracted from each side of an inequality.
Slide 2.8-10
Use the addition property of inequality. (cont’d)
Because an inequality has many solutions, we cannot check all of them by
substitutions as we did with the single solution of an equation. Thus, to
check the solutions of an inequality, first substitute into the equation the
boundary point of the interval and another number from within the
interval to test that they both result in true statements. Next, substitute
any number outside the interval to be sure it gives a false statement.
Slide 2.8-11
CLASSROOM
EXAMPLE 2
Using the Addition Property of Inequality
Solve the inequality, and graph the solution set.
Solution:
1  8x  7 x  2
1 8x  7 x  7 x  2  7x
1  x 1  2 1
x
(,3)
Slide 2.8-12
Objective 3
Use the multiplication property of
inequality.
Slide 2.8-13
Use the multiplication property of inequality.
The addition property of inequality cannot be used to solve an inequality
such as 4x ≥ 28. This inequality requires the multiplication.
Multiply each side of the inequality 3 < 7 by the positive number 2.
2  3  2  7 
6 
True
Now multiply by each side of 3 < 7 by the negative number −5.
5  3  5  7 
15  
False
To get a true statement when multiplying each side by −5, we must reverse
the direction of the inequality symbol.
5  3  5  7 
15  
True
Slide 2.8-14
Use the multiplication property of inequality. (cont’d)
Multiplication Property of Inequality
If A, B, and C, with C ≠ 0,
1. if C is positive, then the inequalities
A  B and AC  BC
have exactly the same solutions;
2. if C is negative, then the inequalities
A  B and AC  BC
have exactly the same solutions.
That is, each of an inequality may be multiplied by the same positive number
without changing the solutions. If the multiplier is negative, we must
reverse the direction of the inequality symbol.
As with the multiplication property of inequality, the same nonzero number
may be divided into each side of an inequality.
Slide 2.8-15
CLASSROOM
EXAMPLE 3
Using the Multiplication Property of Inequality
Solve the inequality, and graph the solution set.
r  12
Solution:
r 

2
2
r 6
(, 6)
Slide 2.8-16
Objective 4
Solve linear inequalities by using both
properties of inequality.
Slide 2.8-17
Solve linear inequalities by using both properties of inequality.
Solving a Linear Inequality
Step 1: Simplify each side separately. Use the distributive
property to clear parentheses and combine like terms on
side as needed.
each
Step 2: Isolate the variable terms on one side. Use the addition
property of inequality to get all terms with variables on one
of the inequality and all numbers on the other side.
side
Step 3: Isolate the variable. Use the multiplication property of
inequality to change the inequality to the form “variable < k”
or “variable > k,” where k is a number.
Remember: Reverse the direction of the inequality symbol only when
multiplying or dividing each side of an inequality by a negative number..
Slide 2.8-18
CLASSROOM
EXAMPLE 4
Solving a Linear Inequality
Solve the inequality, and then graph the solution set.
5x  x  2  7 x  5
Solution:
4x  2  7 x  7 x  5  7 x
3x  2  2  5  2
3 x 7

3 3
7
x
3
7 
 ,
3 
Slide 2.8-19
CLASSROOM
EXAMPLE 5
Solving a Linear Inequality
Solve the inequality, and graph the solution set.
4  x 1  3x  15   2 x  1
Solution:
4x  4  3x  2x    x  2x
3x  4  4    4
3 x 12

3
3
x  4
 4,  
Slide 2.8-20
CLASSROOM
EXAMPLE 6
Solving a Linear Inequality with Fractions
Solve and graph the solution set.
1
3
( x  3)  2  ( x  8)
4
4
1

3

Solution: 4
 4 ( x  3)  2   4  4 ( x  8) 
1

3

4  ( x  3)   4(2)  4  ( x  8) 
4

4

x  3  8  3( x  8)
x  3  8  3x  24
x 11  3x  24
x 11 11  3x  24 11
Multiply by 4.
Distributive
property.
Multiply.
Distributive
property.
Subtract 11.
Slide 2.8- 20
CLASSROOM
EXAMPLE 6
Solving a Linear Inequality with Fractions (cont’d)
x 11 11  3x  24 11
x  3 x  13
x  3x  3x  3x 13
2x  13
Reverse the
inequality symbol
when dividing by a
negative number.
2 x 13

2 2
13
x
2
Subtract 11.
Subtract 3m.
Divide 2.
The solution set is the
interval [−13/2, ).
[
Slide 2.8- 21
Objective 5
Solve applied problems by using
inequalities.
Slide 2.8-23
Solve applied problems by using inequalities.
Inequalities can be used to solve applied problems involving phrases that
suggest inequality. The table gives some of the more common such phrases,
along with examples and translations.
In general, to find the average of n numbers, add the numbers and divide
by n. We use the same six problem-solving steps from Section 2.4, changing
Step 3 to “Write an inequality.”, instead of “Write an equation.”
Do not confuse statements such as “5 is more than a number” with phrases
like “5 more than a number.” The first of these is expressed as 5 > x, while
the second is expressed as x + 5 or 5 + x.
Slide 2.8-24
CLASSROOM
EXAMPLE 7
Using a Linear Inequality to Solve a Rental Problem
A rental company charges $5 to rent a leaf blower, plus $1.75 per hr. Marge
Ruhberg can spend no more than $26 to blow leaves from her driveway and
pool deck. What is the maximum amount of time she can use the rented leaf
blower?
Solution:
Step 1
Read the problem again. What is to be found?
The maximum time Marge can afford to rent the blower.
What is given?
The flat rate to rent the leaf blower, the additional
hourly charge to rent the leaf blower, and the
maximum amount that Marge can spend.
Step 2
Assign a variable.
Let h = the number of hours she can rent the blower.
Slide 2.5- 24
CLASSROOM
EXAMPLE 7
Using a Linear Inequality to Solve a Rental Problem (cont’d)
Step 3 Write an inequality.
She must pay $5, plus $1.75 per hour for h hours
and no more than $26.
Cost of
is no
renting
more than
26
5 + 1.75h
Step 4 Solve.
≤
1.75h ≤ 21
h ≤ 12
26
Subtract 5.
Divide by 1.75.
Step 5 State the answer.
She can use the leaf blower from a maximum of 12
hours.
Step 6 Check.
If she uses the leaf blower for 12 hr, she will spend
5 + 1.75(12) = 26 dollars, the maximum.
Slide 2.5- 25
CLASSROOM
EXAMPLE 8
Finding an Average Test Score
Maggie has scores of 98, 86, and 88 on her first three tests in
algebra. If she wants an average of at least 90 after her fourth test,
what score must she make on that test?
Solution:
Let x = Maggie’s fourth test score.
98  86  88  x
 90
4
272  x
   4 
 4
4
  x  272  360  272
Maggie must get greater than or
equal to an 88.
x  88
Slide 2.8-27
Objective 6
Solve linear inequalities with three
parts.
Slide 2.8-28
Solve linear inequalities with three parts.
Inequalities that say the one number is between two other numbers are
three-part inequalities.
For example,
says
that 5 is between −3 and 7.
3    
For some applications, it is necessary to work with a three-part inequality
such as
3 x2
where x +2 is between 3 and 8. To solve this inequality, we subtract 2 from
each of the three parts of the inequality.
3  2  x  2  2   2
1 x  
Slide 2.8-29
Solve linear inequalities with three parts. (cont’d)
The idea is to get the inequality in the form
a number < x < another number,
using “is less than.” The solution set can then easily be graphed.
When inequalities have three parts, the order of the parts is important. It would be
wrong to write an inequality as 8 < x + 2 < 3, since this would imply 8 < 3, a false
statement. In general, three-part inequalities are written so that the symbols
point in the same direction and both point toward the lesser number.
Slide 2.8-30
CLASSROOM
EXAMPLE 9
Solving Three-Part Inequalities
Write the inequality in interval notation, and graph the interval.
2 x4
Solution:
 2, 4
Slide 2.8-31
CLASSROOM
EXAMPLE 10
Solving Three-Part Inequalities
Solve the inequality, and graph the solution set.
2  3x 1  8
Solution:
2 1  3x 1 1  8 1
3 3x 9


3 3 3
1 x  3
1,3
Remember to work with all three parts of the inequality.
Slide 2.8-32