Exercise 3 Page 191

We are asked to find and graph the solution set for all possible values of x in the given inequality. Let's begin by isolating the absolute value in the inequality. |6x-12|+6 < 30 ⇒ |6x-12| < 24 Now, we will create a compound inequality by removing the absolute value. In this case, the solution set is any number less than 24 away from the midpoint in the positive direction and any number less than 24 away from the midpoint in the negative direction. Absolute Value Inequality:& |6x-12| < 24 Compound Inequality:& -24 < 6x-12< 24

We can split this compound inequality into two cases, one where 6x-12 is greater than -24 and one where 6x-12 is less than 24. 6x-12 >-24 and 6x-12 < 24 Let's isolate x in both of these cases before graphing the solution set.

Case 1

We can now focus on the first inequality.

6x-12<24

AddIneq

LHS+12

6x<36

DivIneq

.LHS /6.<.RHS /6.

x<6

This inequality tells us that all values less than 6 will satisfy the inequality.

Case 2

Let's now move on to the second inequality.

6x-12> -24

AddIneq

LHS+12>RHS+12

6x>-12

DivIneq

.LHS /6.>.RHS /6.

x>-2

RearrangeIneq

Rearrange inequality

-2< x

This inequality tells us that all values greater than -2 will satisfy the inequality.

Solution Set

The solution to this type of compound inequality is the overlap of the solution sets. Let's recombine our cases back into one compound inequality. First Solution Set:& x < 6 Second Solution Set:& -2 < x Intersecting Solution Set:& -2 < x < 6

Graph

The graph of this inequality includes all values from -2 to 6, not inclusive. We show this by using open circles on the endpoints.

Hints

Check the answer

Practice exercises

Asses your skill

Case 1

Case 2

Solution Set

Graph