# HOW TO SOLVE ABSOLUTE VALUE INEQUALITIES ON A NUMBER LINE

Example 1 :

Solve for x :

2|x + 1| - 6 < 7

Graph the solution set in a number line

Solution :

2|x + 1| - 6  <  7

2|x + 1|  <  13

Divide by 2 on both sides

|x + 1|  <  13/2

Now we may split this into two branches as shown below.

 x + 1  <  13/2Subtract 1 from each side. x  <  (13/2) - 1 x  <  11/2 (x + 1)  >  -13/2Subtract 1 from each side. x  >  (-13/2) - 1 x  >  -15/2

So, the solution is

-15/2 < x < 11/2

By graphing the above solution set, we get

By representing the above graph in solution set, we get

(-∞, -15/2) U (11/2, ∞)

Example 2 :

Solve for x :

(1/5)|10x - 2|  <  1

Solution :

(1/5) |10x - 2|  <  1

Multiply each side by 5.

|10x - 2|  <  5

 10x - 2  >  -5Add 2 to each side.10x  >  -3Divide each side by 10. x  >  -3/10 10x - 2  <  5Add 2 to each side.10x  <  7Divide each side by 10. x  <  7/10

So, the solution is

-3/10 < x < 7/10.

Example 3 :

Solve for x :

|5x - 12| < -2

Solution :

|5x - 12| < -2

In the right side, we have numerical value -2 and in the left side, we have a absolute value function.

By applying any values to x, we get only positive values on the right side. So the given statement will never become true for any value of x.

Hence, there is no solution for the given inequality.

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