Solving Absolute Value Equations and Inequalities

By Ellie Silverman

Solving Absolute Value Equations

2x + 3 = 7

2x + 3 = 7

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.

These are absolute value signs

4-2 0-1 1 2 3-4 -3

1 2 3 4 1 2 3 4

2x + 3 = 7To 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

2x + 3 = 7

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

2x + 3 = -7

2x = 4 2x = -102 2 22x = 2 x = -5

-3 -3 -3 -3

Solving Absolute Value Inequalities

2x + 3 < 72x + 3 ≤ 7

2x + 3 > 72x + 3 ≥ 7

Some things to remember when solving inequalities:

* < or > = open dot on number line

* 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 ≥ = closed dot on number line

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

Less thand

2x + 3 < 7

2x + 3 > -7an

d

Now that you set 2x+3 less than 7 and -7, you solve.

2x + 3 < 7

and

2x + 3 > -7-3 -3 -3 -3

2x < 4 2x > -10

x < 2 x > -52 2 2 2

4-2 0-1 1 2 3-4 -3

Now it’s time to graph your solutions!

5-5-6

X < 2 AND X > -5

If the answer looked like this:

X ≤ 2 AND X ≥ -5 Then the circles on the graph would be closed

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

greator

2x + 3 > 7

2x + 3 < -7or

Now that you set 2x+3 greater than 7 and less than -7, you solve.

2x + 3 > 7

and

2x + 3 < -7-3 -3 -3 -3

2x > 4 2x < -10

x > 2 x < -52 2 2 2

4-2 0-1 1 2 3-4 -3

Now it’s time to graph your solutions!

X > 2 OR x < -5

5-5-6

If the answer looked like this:

Then the circles on the graph would be closed

X ≥ 2 OR x ≤ -5

More PracticeSolve and graph each inequality:

1.) 8b + 9 < 63

4.) -8x - 10 < 58

2.) 4 + 5x > 24

3.) 5 - 8x < 43

5.) -7r ≤ 7

6.) 8b + 9 < 63

7.) -5p ≥ 20

3.) 1 + x < 8

Just a Funny Math Video

http://www.youtube.com/watch?v=6cAs1YBELmA