If x is an integer that satisfies both |4x - 5| \le 13 and \frac{2x - 1}{-3} \le -1, what … — GRE General Test

If $x$ is an integer that satisfies both $|4x - 5| \le 13$ and $\frac{2x - 1}{-3} \le -1$ , what is the sum of all possible values of $x$ ?

Solving the absolute value inequality

|4x - 5| \le 13

yields

-13 \le 4x - 5 \le 13

, which simplifies to

-2 \le x \le 4.5

. Next, solving

\frac{2x - 1}{-3} \le -1

requires multiplying both sides by

-3

and reversing the inequality sign, giving

2x - 1 \ge 3

, which simplifies to

x \ge 2

. Combining

-2 \le x \le 4.5

and

x \ge 2

for integer

x

gives the set

\{2, 3, 4\}

. The sum of these values is

2 + 3 + 4 = 9

Solve the absolute value inequality

|4x - 5| \le 13

-13 \le 4x - 5 \le 13 \implies -8 \le 4x \le 18 \implies -2 \le x \le 4.5

An absolute value inequality

|u| \le k

unfolds into the compound inequality

-k \le u \le k

Solve the linear inequality

\frac{2x - 1}{-3} \le -1

2x - 1 \ge 3 \implies 2x \ge 4 \implies x \ge 2

Multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality sign.

Find the intersection of the two solution sets for integer

x

x \in \{2, 3, 4\}

The integers satisfying both

-2 \le x \le 4.5

and

x \ge 2

are 2, 3, and 4.

Calculate the sum of the possible integer values

2 + 3 + 4 = 9

Summing the valid integer solutions yields the final requested answer.

Linear Inequalities and Absolute Value

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