1. Section 4 Solving Inequalities
Chapter 1
Section 4
Solving Inequalities

2. Solving Inequalities State whether each inequality is true or false.
ALGEBRA 2 LESSON 1-4
(For help, go to Lessons 1-1 and 1-3.)
State whether each inequality is true or false.
1. 5 < 12
2. 5 < –12
–12
> –
< –
< –
> –
Solve each equation.
7. 3x + 3 = 2x – 3
8. 5x = 9(x – 8) + 12

3. Solving Inequalities Solutions 1. 5 < 12, true 3. 5 12, false
ALGEBRA 2 LESSON 1-4
Solutions
1. 5 < 12, true
, false
, true
7. 3x + 3 = 2x – 3
3x – 2x = –3 – 3 x = –6
2. 5 < –12, false
–12, false
, true
8. 5x = 9(x – 8) + 12
5x = 9x – –4x = –60 x = 15
< –
> –
> –
< –

4. Inequalities The solutions include more than one number
Ex: 2 < x ;values that x could be include 3, 7, 45…
All of the rules for solving equations apply to inequalities, with one added:
If you multiply or divide by a NEGATIVE you must FLIP the sign. (< becomes > and > becomes <)
When graphing on a number line:
Open dot for < or >
Closed (solid) dot for ≤ or ≥
The shading should be easy to see (a slightly elevated line is ok) --- see examples

5. Solving Inequalities –2x < 3(x – 5)
ALGEBRA 2 LESSON 1-4
Solve –2x < 3(x – 5). Graph the solution.
–2x < 3(x – 5)
–2x < 3x – 15 Distributive Property
–5x < –15 Subtract 3x from both sides.
x > 3 Divide each side by –5 and reverse the inequality.

6. Try These Problems Solve each inequality. Graph the solution.
3x – 6 < 27
3x < x < 11
12 ≥ 2(3n + 1) + 22
12 ≥ 6n ≥ 6n ≥ 6n -2 ≥ n

7. Solve 7x ≥ 7(2 + x). Graph the solution.
7x ≥ x Distributive Property
0 ≥ 14 Subtract 7x from both sides.
The last inequality is always false, so 7x ≥ 7(2 + x) is always false. It has no solution.

8. Try These Problems Solve. Graph the solution. 2x < 2(x + 1) + 3
2x < 2x x < 2x < 5 All Real Numbers
4(x – 3) + 7 ≥ 4x + 1
4x – ≥ 4x x + 12 ≥ 4x ≥ 1 No Solution

9. Solving Inequalities
A real estate agent earns a salary of $2000 per month plus 4% of the sales. What must the sales be if the salesperson is to have a monthly income of at least $5000?
Relate: $ % of sales $5000
> –
Define: Let x = sales (in dollars).
Write: x
> –
0.04x Subtract 2000 from each side.
> –
x 75,000 Divide each side by 0.04.
> –
The sales must be greater than or equal to $75,000.

10. Try This Problem
A salesperson earns a salary of $700 per month plus 2% of the sales. What must the sales be if the salesperson is to have a monthly income of at least $1800?
x ≥ 1800
.02x ≥ 1100
x ≥ 55000
The sales must be at least $55,000.

11. Compound Inequalities
Compound Inequality – a pair of inequalities joined by and or or
Ex: -1 < x and x ≤ 3 which can be written as < x ≤ 3
x < -1 or x ≥ 3
For and statements the value must satisfy both inequalities
For or statements the value must satisfy one of the inequalities

12. And Inequalities
Graph the solution of 3x – 1 > -28 and 2x + 7 < 19.
3x > -27 and 2x < 12
x > -9 and x < 6
Graph the solution of -8 < 3x + 1 <19
-9 < 3x < 18
-3 < x < 6

13. Or Inequalities
ALGEBRA 2 LESSON 1-4
Graph the solution of 3x + 9 < –3 or –2x + 1 < 5.
3x + 9 < –3 or –2x + 1 < 5
3x < – –2x < 4
x < –4 or x > –2

14. Try These Problems
Graph the solution of 2x > x + 6 and x – 7 < 2
x > 6 and x < 9
Graph the solution of x – 1 < 3 or x + 3 > 8
x < 4 or x > 11

15. Homework
Practice 1.4 All omit 23