If x satisfies the inequality |3 - 2x| > 9, which of the following could be the value of x… — GRE General Test

If $x$ satisfies the inequality $|3 - 2x| > 9$ , which of the following could be the value of $x$ ? Select all that apply.

The values

-5

and

8

satisfy the inequality.

The absolute value inequality

|3 - 2x| > 9

splits into

3 - 2x > 9

3 - 2x < -9

. Solving these yields

x < -3

x > 6

. The value

-5

is less than

-3

and the value

8

is greater than

6

, so both are valid solutions.

Set up the two separate linear inequalities based on the definition of absolute value.

3 - 2x > 9

3 - 2x < -9

An absolute value expression

|u| > c

(where

c > 0

) breaks into two disjoint cases:

u > c

u < -c

Solve the first case:

3 - 2x > 9

-2x > 6 \implies x < -3

Subtract 3 from both sides to get

-2x > 6

, then divide by

-2

and reverse the inequality sign.

Solve the second case:

3 - 2x < -9

-2x < -12 \implies x > 6

Subtract 3 from both sides to get

-2x < -12

, then divide by

-2

and reverse the inequality sign.

Combine the solution sets and evaluate the given options.

The valid solution set is

x < -3

x > 6

. Among the choices,

-5

(since

-5 < -3

) and

8

(since

8 > 6

) fall into the solution set.

Values between

-3

and

6

, inclusive, do not satisfy the original inequality.

Solving absolute value inequalities of the form

|u| > c

by splitting into compound inequalities and reversing the direction of inequality signs when dividing by negative numbers.

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