1. 4.1 Solving Linear Inequalities
10/11/2013

2. Review of Inequality Signs
> greater than
< less than
greater than or equal
less than or equal

3. > > > > < < Consider 4>2 Add 2 to both sides.
4 > 2
Subtract 2 from both sides.
4 > 2
−2 −2
>
>
Multiply both sides by 2.
4 > 2
×2 ×2
Divide both sides by 2.
4 > 2
÷2 ÷2
>
>
Divide both sides by -2.
4 > 2
÷−2 ÷−2
− −1
Multiply both sides by -2.
4 >
×−2 × −2
− −4
When did you change the direction of the inequality symbol?
<
<

4. Rule of Inequality When multiplying or dividing both sides of an inequality by a negative number, reverse the inequality symbol.

5. How to graph the solutions
> Graph any number greater than. . .
open circle, line to the right
< Graph any number less than. . .
open circle, line to the left
Graph any number greater than or equal to. . .
closed circle, line to the right
Graph any number less than or equal to. . .
closed circle, line to the left

6. Solve the inequality and graph the solution.
Example 1
Inequality with a Variable on One Side
Solve the inequality and graph the solution. a.
> x – 6 b. – ≥ 5y 13 +
SOLUTION a.
> x – 6 b. – ≥ 5y 13 + – ≥ 5y 15
+ 4
+ 4
> x 2 –
Reverse symbol y ≤ 3
Open circle, line to the right.
Closed circle, line to the left.

7. > Example 2 Solve . Graph the solution. < 7 4x 1 2x – SOLUTION
Inequality with a Variable on Both Sides
Solve Graph the solution.
< x x –
SOLUTION
< x x –
Open circle, line to the right.
+2x x
< x – 1
< 2x – 6
Reverse symbol
> x 3 7

8. Solve the inequality. Then graph your solution.
Checkpoint
Solve an Inequality
Solve the inequality. Then graph your solution.
< x 8 + 1.
ANSWER
< x 5 2 1 3 4 6 5 ≤ x – 2.
ANSWER ≥ x 1 – 2

9. Solve the inequality. Then graph your solution.
Checkpoint
Solve an Inequality
Solve the inequality. Then graph your solution. x
> 2 3
ANSWER 1 2
>
2x 1 – 3. – x
> 2x 4. 3 x
>
ANSWER 2 1 4 5 6

10. You can buy up to 24 ride tickets.
Example 3
Word Problem
Amusement Park Admission to an amusement park costs $9 and each ride ticket costs $1.50.
Write an equation for the amount you’ll spend for any given number of ride tickets you buy.
If you only have $45, up to how many ride tickets can you buy?
9+1.50𝑥≤45
− − 9
1.50𝑥 ≤ 36
𝑥 ≤ 24
You can buy up to 24 ride tickets.
1 ticket:9+1.50=𝐶($10.50)
2 tickets: (2) =𝐶($12)
3 tickets: (3) = 𝐶(13.50)
4 tickets: (4) = 𝐶($15)
x tickets: (𝑥) =𝐶 10

11. Classwork Worksheet 4.1 Front page only

12. 2 simple inequalities joined by the word “and” or the word “or”.
COMPOUND INEQUALITY
2 simple inequalities joined by the word “and” or the word “or”.
Ex: AND
All real numbers greater than or equal to -2 and less than 1.
Ex: OR
All real numbers less than -1 or greater than or equal to 2
𝑥≥−2
𝑥<1
and
is the same as
-2≤ x < 1
Note: The “shaded” part is in the middle of the 2 circled numbers.
x < -1 or x ≥ 2
Note: The “shaded” parts go in opposite direction from 2 circled numbers.

13. 3𝑥+2 < 8 2𝑥−9>3 −2 −2 +9 +9 3𝑥 < 6 2𝑥 > 12 3 3 2 2
Example 4
Solve an “Or” Compound Inequality
Solve or 3x + 2 8
< 2x 9 3. –
>
SOLUTION
Solve each part separately.
3𝑥+2 < 8
−2 −2
3𝑥 < 6
𝑥 < 2
2𝑥−9>3
2𝑥 > 12
𝑥 > 6 or 13

14. −𝟏≤𝟑𝒙+𝟖≤𝟖 −8 −8 −8 −9 ≤ 3𝑥 ≤ 0 3 3 3 −3 ≤ 𝑥 ≤ 0 Example 5
Solve an “And” Compound Inequality
Solve −𝟏≤𝟑𝒙+𝟖≤𝟖
SOLUTION
−𝟏≤𝟑𝒙+𝟖≤𝟖
− − −8
−9 ≤ 3𝑥 ≤ 0
−3 ≤ 𝑥 ≤ 0 14

15. Solve the inequality. Then graph your solution.
Checkpoint
Solve Compound Inequalities
Solve the inequality. Then graph your solution. 5. 4 x
< + 5 7
ANSWER x 2
< 1 –

16. . Checkpoint Solve the inequality. Then graph your solution. 6. x + 3
Solve Compound Inequalities
Solve the inequality. Then graph your solution. 6. x + 3
4 or ≤ 6 – ≥ 1
ANSWER x
1 or ≤ 5 ≥ 4 2 . 7. 2x 6
< – x
4 or
>
ANSWER x
4 or –
< 3
> 6 4 2

17. Homework Worksheet 4.1 Back page