What is the minimum integer value of x that satisfies the inequality |2x - 7| \le 5? — GRE General Test

What is the minimum integer value of $x$ that satisfies the inequality $|2x - 7| \le 5$ ?

Answer: 1

The minimum integer value of

x

that satisfies the inequality is 1.

To solve

|2x - 7| \le 5

, write it as the compound inequality

-5 \le 2x - 7 \le 5

. Adding 7 across all parts gives

2 \le 2x \le 12

. Dividing by 2 yields

1 \le x \le 6

. The integer solutions are 1, 2, 3, 4, 5, and 6. The minimum integer among these is 1.

Convert the absolute value inequality into a compound inequality.

-5 \le 2x - 7 \le 5

An inequality of the form

|u| \le a

(where

a \ge 0

) is equivalent to

-a \le u \le a

Add 7 to all three parts of the inequality.

2 \le 2x \le 12

Adding a constant to an inequality preserves the direction of the inequality signs.

Divide all three parts by 2.

1 \le x \le 6

Dividing by a positive constant isolates

x

without reversing the inequality signs.

Determine the minimum integer within the solution set

[1, 6]

The solution set contains integers

\{1, 2, 3, 4, 5, 6\}

, making 1 the smallest integer value.

Solving absolute value inequalities using equivalent compound linear inequalities

Question