If x is a real number that satisfies the inequality \left|\frac{5 - 2x}{3}\right| \le 3, w… — GRE General Test

If $x$ is a real number that satisfies the inequality $\left|\frac{5 - 2x}{3}\right| \le 3$ , what is the minimum possible value of $4 - 3x$ ?

-17

To find the minimum possible value of

4 - 3x

, we first solve the inequality

\left|\frac{5 - 2x}{3}\right| \le 3

. Multiplying by

3

gives

|5 - 2x| \le 9

. This unfolds into the compound inequality

-9 \le 5 - 2x \le 9

. Subtracting

5

gives

-14 \le -2x \le 4

. Dividing all parts by

-2

requires flipping the inequality signs, resulting in

7 \ge x \ge -2

, or

-2 \le x \le 7

. Because

4 - 3x

decreases as

x

increases, the expression reaches its minimum when

x

is at its maximum value of

7

. Substituting

x = 7

yields

4 - 3(7) = -17

Clear the denominator from the absolute value inequality

|5 - 2x| \le 9

Multiplying both sides of the inequality by the positive number 3 preserves the inequality direction.

Express the absolute value inequality as a compound inequality

-9 \le 5 - 2x \le 9

For any non-negative constant

c

|u| \le c

is equivalent to

-c \le u \le c

Isolate the variable term by subtracting 5 from all parts

-14 \le -2x \le 4

Subtracting a constant from all parts of a compound inequality maintains the inequality relationships.

Divide by -2 and reverse the inequality signs

7 \ge x \ge -2 \implies -2 \le x \le 7

Dividing an inequality by a negative number reverses the direction of the inequality signs.

Determine which bound of

x

minimizes

4 - 3x

and evaluate

Minimum value

= 4 - 3(7) = -17

The expression

4 - 3x

has a negative coefficient for

x

, making it a decreasing function. Therefore, the minimum value of

4 - 3x

occurs when

x

takes its maximum possible value (

x = 7

Linear Inequalities and Absolute Value

Question