How many integer values of x satisfy the inequality |x - 3| \leq 4? — GMAT

How many integer values of $x$ satisfy the inequality $|x - 3| \leq 4$ ?

Answer: 9

There are 9 integer values of

x

that satisfy the inequality.

Unfolding

|x - 3| \leq 4

yields

-4 \leq x - 3 \leq 4

. Adding 3 across all terms gives

-1 \leq x \leq 7

. The integer values satisfying this range are

-1, 0, 1, 2, 3, 4, 5, 6, 7

, which totals 9 values.

Convert the absolute value inequality into a compound linear inequality.

-4 \leq x - 3 \leq 4

The absolute value inequality

|k| \leq c

(for

c \geq 0

) is equivalent to

-c \leq k \leq c

Isolate

x

by adding 3 across all parts of the inequality.

-1 \leq x \leq 7

Adding the same constant to all parts of an inequality maintains the inequality relationships.

Count the total number of integer values within the inclusive range

[-1, 7]

The number of integers in an inclusive range

[a, b]

is calculated as

b - a + 1

, which gives

7 - (-1) + 1 = 9

Solving absolute value inequalities and counting integer solutions within a bounded interval

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