The document covers solving linear and compound inequalities, including how to manipulate formulas to isolate variables. It provides guided practice examples for graphing inequality solutions and solving for specific variables. Key concepts include understanding the definitions of linear and compound inequalities, as well as techniques for visualizing and solving these mathematical expressions.

1. 1.6: Solve Linear
Inequalities
I can:
Solve a linear and/or compound inequality and graph
the solution set.
Manipulate a formula to solve for a particular variable.

2. Key Vocabulary
• Linear Inequality
• Compound Inequality
• Equivalent Inequalities

3. Linear Inequality
• Can be written in one of the following forms where a and b are
real numbers and a does not equal 0.

5. Compound Inequality
• Consists of two simple inequalities joined by “and” or “or”

7. GUIDED PRACTICE for Examples 1 and 2
Graph the inequality.
1. x > – 5
The solutions are all real numbers greater than 5.
An open dot is used in the graph to indicate – 5 is
not a solution.

8. GUIDED PRACTICE for Examples 1 and 2
Graph the inequality.
2. x ≤ 3
The solutions are all real numbers less than or
equal to 3.
A closed dot is used in the graph to indicate 2 is a
solution.

9. GUIDED PRACTICE for Examples 1 and 2
Graph the inequality.
3. – 3 ≤ x < 1
The solutions are all real numbers that are greater
than or equalt to – 3 and less than 1.

10. GUIDED PRACTICE for Examples 1 and 2
Graph the inequality.
4. x < 1 or x ≥ 2
The solutions are all real numbers that are less than 1
or greater than or equal to 2.

11. Why?
• Near the end of the semester or school year, I have students
ask me what scores they must get on a final in order to earn
a particular grade. You can use linear inequalities to
answer this question.
• You budgeted x amount of dollars for movies. You have
already spend $50 of the amount. You can use inequalities
to determine staying under budget or reaching your
budgeted target.

12. Equivalent Inequalities

16. GUIDED PRACTICE for Examples 3 and 4
Solve the inequality. Then graph the solution.
5. 4x + 9 < 25
4x + 9 < 25 Write original inequality.
4x < 16 Subtract 9 from each side.
x < 4 Divide each side by 4.
6. 1 – 3x ≥ – 14
1 – 3x ≥ – 14 Write original inequality.
– 3x ≥ – 15 Subtract –1 from each side.
x ≤ 5

17. GUIDED PRACTICE for Examples 3 and 4
Solve the inequality. Then graph the solution.
7. 5x – 7 ≤ 6x
8. 3 – x > x – 9
5x – 7 ≤ 6x Write original inequality.
x > – 7 Subtract 5x from each side.
3 – x > x – 9 Write original inequality.
3 – 2x > – 9
– 2x > – 12
x < 6
Subtract x from each side.
Subtract 3 from each side.
Divide each side by –2 and reverse.

19. EXAMPLE 6 Solve an “or” compound inequality
Solve 3x + 5 ≤
11 or 5x – 7 ≥ 23
. Then graph the solution.
SOLUTION
A solution of this compound inequality is a solution of
either of its parts.
First Inequality Second Inequality
3x + 5 ≤
11 3x ≤
6
x ≤
2
Write first inequality.
Subtract 5 from each side.
Divide each side by 3.
5x – 7 ≥
23 5x ≥
30
x ≥ 6
Write second inequality.
Add 7 to each side.
Divide each side by 5.

20. GUIDED PRACTICE for Examples 5,6, and 7
Solve the inequality. Then graph the solution.
9. – 1 < 2x + 7 < 19
– 1 < 2x + 7 < 19 Write original inequality.
Subtract 7 from each expression.
Simplify.
Divide each expression by 2.
– 1– 7 < 2x + 7 – 7 < 19 – 7
– 8 < 2x < 12
– 4 < x < 6
ANSWER
The solutions are all real numbers greater than – 4
and less than 6.

21. GUIDED PRACTICE for Examples 5,6 and 7
Solve the inequality. Then graph the solution.
10. – 8 ≤
– x – 5 ≤
6
– 8 ≤– x – 5 ≤
6
– 8+5 ≤– x – 5 + 5
≤ 6 + 5
– 3 ≤– x ≤ 11
– 11 ≤ x ≤
3
Write original inequality.
Add 5 to each expression.
Simplify.
The solutions are all real numbers greater than and
equal to – 11 and less than and equal to 3.
ANSWER

22. GUIDED PRACTICE for Examples 5,6 and 7
Solve the inequality. Then graph the solution.
11. x + 4 ≤
9 or x – 3 ≥ 7
SOLUTION
A solution of this compound inequality is a solution of
either of its parts.
First Inequality Second Inequality
x ≤
5
Write first inequality.
Subtract 4 from each side. x ≥
10
Write second inequality.
Add 3 to each side.
x + 4 ≤
9
x – 3 ≥ 7

23. GUIDED PRACTICE for Examples 5,6 and 7
ANSWER
The graph is shown below. The solutions are all real
numbers.
less than or equal to 5 or greater than or equal to 10.

24. GUIDED PRACTICE for Examples 5,6 and 7
Solve the inequality. Then graph the solution.
12. 3x – 1 < – 1 or 2x + 5 ≥
11
SOLUTION
A solution of this compound inequality is a solution of
either of its parts.
First Inequality Second Inequality
3x ≤
0
x ≤
0
Write first inequality.
Add 1 each side .
Divide each side by 3.
2x + 5 ≥
11 2x ≥ 6
x ≥ 3
Write second inequality.
Subtract 5 from each side
Divide each side by 5.
3x – 1< – 1

25. GUIDED PRACTICE for Examples 5,6 and 7
less than 0 or greater than or equal to 3.
ANSWER
The graph is shown below. The solutions are all real
numbers.