How many integer values of x satisfy the inequality \big||x + 1| - 4\big| \leq 3 - |x|? — GMAT

How many integer values of $x$ satisfy the inequality $\big||x + 1| - 4\big| \leq 3 - |x|$ ?

The correct answer is 4. The inequality requires

3 - |x| \geq 0

, restricting candidate solutions to integers in the range

[-3, 3]

. Testing each integer directly reveals that only

x = 0, 1, 2, 3

satisfy the inequality, yielding a total of 4 valid integers.

Establish the domain restriction from the non-negativity of the absolute value.

Since the left-hand side

\big||x + 1| - 4\big| \geq 0

for all real numbers, the right-hand side must also be non-negative:

3 - |x| \geq 0 \implies |x| \leq 3 \implies -3 \leq x \leq 3

An absolute value expression cannot be less than a negative number.

Test non-negative integer candidates in the domain

[-3, 3]

For

x = 3

\big|4 - 4\big| = 0 \leq 3 - 3 = 0

(True);
For

x = 2

\big|3 - 4\big| = 1 \leq 3 - 2 = 1

(True);
For

x = 1

\big|2 - 4\big| = 2 \leq 3 - 1 = 2

(True);
For

x = 0

\big|1 - 4\big| = 3 \leq 3 - 0 = 3

(True).

Substituting non-negative integers verifies that

x \in \{0, 1, 2, 3\}

satisfy the inequality.

Test negative integer candidates in the domain

[-3, 3]

For

x = -1

\big|0 - 4\big| = 4 \leq 3 - 1 = 2

(False);
For

x = -2

\big|-1 - 4\big| = 5 \leq 3 - 2 = 1

(False);
For

x = -3

\big|-2 - 4\big| = 6 \leq 3 - 3 = 0

(False).

Evaluating negative integers shows no negative integer satisfies the inequality.

Count the total number of valid integer solutions.

The valid integer solutions are

x = 0, 1, 2, 3

, giving a total count of 4.

Combining all valid cases yields exactly 4 integer solutions.

Solving nested absolute value inequalities using domain constraints and piecewise sign analysis.

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