1. 2
Chapter
Chapter 2
Equations, Inequalities and Problem Solving

2. Solving Linear Inequalities
Section
2.8
Solving Linear Inequalities

3. Objective 1
Define Linear Inequality in One Variable, Graphing Solution Sets on a Number Line, and Use Interval Notation

4. Linear Inequalities
An inequality is a statement that contains of the symbols: < , >, ≤ or ≥.
Equations Inequalities
x = 3 x > 3
12 = 7 – 3y 12 ≤ 7 – 3y

5. Graphing Solutions
Graphing solutions to linear inequalities in one variable
Use a number line
Use a bracket at the endpoint of a interval if you want to include the point
Use a parenthesis at the endpoint if you DO NOT want to include the point
Represents the set {xx  7}
Represents the set {xx > – 4}

6. Solutions to Linear Inequalities
x < 3
Interval notation:
(–∞, 3)
NOT included
–2 < x < 0
Interval notation:
(–2, 0)
Included
–1.5  x  3
Interval notation:
[–1.5, 3)

7. Example
Graph:
In interval notation, this is [ –1, ∞).

8. Solving Linear Inequalities
Objective 2
Solving Linear Inequalities

9. Addition Property of Inequality
If a, b, and c are real numbers, then
a < b and a + c < b + c
are equivalent inequalities.

10. Example
Solve Graph the solution set and write it in interval notation. [

11. Multiplication Property of Inequality
1. If a, b, and c are real numbers, and c is positive, then
a < b and ac < bc
are equivalent inequalities.
2. If a, b, and c are real numbers, and c is negative, then
a < b and ac > bc are equivalent inequalities.

12. Example
Solve Graph the solution set and write it in interval notation. [

13. Solving Linear Inequalities in One Variable
Step 1: Clear the inequality of fractions by multiplying both sides of the inequality by the LCD of all fractions in the inequality.
Step 2: Remove grouping symbols such as parentheses by using the distributive property.
Step 3: Simplify each side of inequality by combining like terms.
Step 4: Write the inequality with variable terms on one side and numbers on the other side by using the addition property of inequality.
Step 5: Get the variable alone by using the multiplication property of inequality.

14. Example
Solve: 3x + 8 ≥ 5. Graph the solution set.

15. Example Solve: 3x + 9 ≥ 5(x – 1). Graph the solution set.
3x + 9 ≥ 5x – Apply the distributive property.
3x – 3x + 9 ≥ 5x – 3x – Subtract 3x from both sides.
9 ≥ 2x – Simplify.
9 + 5 ≥ 2x – Add 5 to both sides.
14 ≥ 2x Simplify.
7 ≥ x Divide both sides by 2.
The solution set [7, ∞)

16. Example
Solve: 7(x – 2) + x > –4(5 – x) – 12. Graph the solution set.
7(x – 2) + x > –4(5 – x) – 12
7x – 14 + x > –20 + 4x – 12 Apply the distributive property.
8x – 14 > 4x – 32 Combine like terms.
8x – 4x – 14 > 4x – 4x – 32 Subtract 4x from both sides.
4x – 14 > –32 Simplify.
4x – > – Add 14 to both sides.
4x > –18 Simplify.
Divide both sides by 4.
The graph of solution set is (–9/2, ∞).

17. Solving Compound Inequalities
Objective 3
Solving Compound Inequalities

18. Compound Inequalities
A compound inequality contains two inequality symbols.
0  4(5 – x) < 8
This means 0  4(5 – x) and 4(5 – x) < 8.
To solve the compound inequality, perform operations simultaneously to all three parts of the inequality (left, middle and right).

19. Example
Graph:

20. Example
Solve the inequality. Graph the solution set and write it in interval notation.
9 < z + 5 < 13
9 < z + 5 < 13
9 – 5 < z + 5 – 5 < 13 – 5 Subtract 5 from all three parts.
4 < z < 8
(4, 8)

21. Example
Solve the inequality. Graph the solution set and write it in interval notation.
–7 < 2p – 3 ≤ 5
–7 < 2p – 3 ≤ 5
–4 < 2p ≤ 8 Add 3 to all three parts.
–2 < p ≤ 4 Divide all three parts by 2.
(–2, 4]

22. Example
Solve the inequality. Graph the solution set and write it in interval notation.
0  20 – 4x < 8
0  20 – 4x < Use the distributive property.
0 – 20  20 – 20 – 4x < 8 – Subtract 20 from each part.
– 20  – 4x < – Simplify each part.
5  x > Divide each part by –4.
Remember that the sign changes direction when you divide by a negative number.
The solution is (3,5].

23. Solving Inequality Applications
Objective 4
Solving Inequality Applications

24. Example
Six times a number, decreased by 2, is at least 10. Find the number.
1. UNDERSTAND
Let x = the unknown number.
“Six times a number” translates to 6x,
“decreased by 2” translates to 6x – 2,
“is at least 10” translates ≥ 10.
Continued

25. Example (cont) 2. TRANSLATE Continued Six times a number 6x decreased–
by 2 2
is at least ≥ 10
Continued

26. Example (cont) 3. SOLVE 4. INTERPRET
6x ≥ Add 2 to both sides.
x ≥ 2 Divide both sides by 6.
4. INTERPRET
Check: Replace “number” in the original statement of the problem with a number that is 2 or greater.
Six times 2, decreased by 2, is at least 10
6(2) – 2 ≥ 10
10 ≥ 10 State: The number is 2.

27. Example
You are having a catered event. You can spend at most $1200. The set up fee is $250 plus $15 per person, find the greatest number of people that can be invited and still stay within the budget. Let x represent the number of people Set up fee + cost per person × number of people ≤ x ≤ 1200
continued

28. Example (cont)
You are having a catered event. You can spend at most $1200. The set up fee is $250 plus $15 per person, find the greatest number of people that can be invited and still stay within the budget.
The number of people who can be invited must be 63 or less to stay within the budget.