1. Solving Absolute Value Equations and Inequalities
By Ellie Silverman

2. Solving Absolute Value Equations
2x + 3 = 7

3. 2x + 3 = 7 This equation means the absolute value of 2x+3 is 7.
These are absolute value signs
This equation means the absolute value of 2x+3 is 7.
Absolute value is a number’s distance from zero. For example, the absolute value of 4 is 4 and the absolute value of -4 is 4 because they are both four ‘spots’ away from zero. 1 2 3 4 1 2 3 4 -4 -3 -2 -1 1 2 3 4

4. 2x + 3 = 7
To solve this equation you have to get rid of the absolute value signs. To do this you have to set 2x+3 equal to 7 and -7.
2x + 3 = 7
2x + 3 = -7

5. Now that you set 2x+3 equal to 7 and -7, you solve.
-3 -3
-3 -3
2x = 4
2x = -10 2 2 2 2
x = -5
x = 2

6. Solving Absolute Value Inequalities
2x < 7
2x + 3 > 7
2x + 3 ≤ 7
2x + 3 ≥ 7

7. Some things to remember when solving inequalities:
* Always remember to flip the sign if you divide by a negative
* Less than means you put and between your solutions and shade between them on the number line
* Greater means you put or between your solutions and shade outwards on the number line
* < or > = open dot on number line
* ≤ or ≥ = closed dot on number line

8. 2x + 3 < 7 2x + 3 < 7 2x + 3 > -7 and
Less thand
2x + 3 < 7
To solve this inequality you have to get rid of the absolute value signs. To do this you have to set 2x+3 less than 7 and greater than -7
2x + 3 < 7
2x + 3 > -7
and

9. Now that you set 2x+3 less than 7 and -7, you solve.
-3 -3
-3 -3
2x < 4
2x > -10 2 2 2 2
x < 2
x > -5

10. X < 2 AND X > -5 X ≤ 2 AND X ≥ -5
Now it’s time to graph your solutions!
X < 2 AND X > -5 -6 -5 -4 -3 -2 -1 1 2 3 4 5
If the answer looked like this:
X ≤ 2 AND X ≥ -5
Then the circles on the graph would be closed

11. 2x + 3 > 7 2x + 3 > 7 2x + 3 < -7 or
greator
2x + 3 > 7
To solve this inequality you have to get rid of the absolute value signs. To do this you have to set 2x+3 greater than 7 and less than -7
2x + 3 > 7
2x + 3 < -7 or

12. Now that you set 2x+3 greater than 7 and less than -7, you solve.
-3 -3
-3 -3
2x > 4
2x < -10 2 2 2 2
x > 2
x < -5

13. X > 2 OR x < -5 X ≥ 2 OR x ≤ -5
Now it’s time to graph your solutions!
X > 2 OR x < -5 -6 -5 -4 -3 -2 -1 1 2 3 4 5
If the answer looked like this:
X ≥ 2 OR x ≤ -5
Then the circles on the graph would be closed

14. Solve and graph each inequality:
More Practice
Solve and graph each inequality:
1.) 8b + 9 < 63
5.) -7r ≤ 7
6.) 8b + 9 < 63
2.) 4 + 5x > 24
3.) 5 - 8x < 43
7.) -5p ≥ 20
4.) -8x < 58
3.) x < 8

15. Just a Funny Math Video