ACT Math: How Many Integers Satisfy |2x − 7| < 9?

The Problem

How many integers $x$ satisfy $|2x - 7| < 9$?

>

F) 7 G) 8 H) 9 J) 10 K) infinitely many

Step 1: Remove the Absolute Value

$|A| < c$ (with $c > 0$) means $-c < A < c$:

$$ -9 < 2x - 7 < 9 $$

Step 2: Isolate x in All Three Parts

Add 7 to every part:

$$ -2 < 2x < 16 $$

Divide every part by 2 (a positive number, so the signs stay as they are):

$$ -1 < x < 8 $$

Step 3: Count the Integers

Both endpoints are excluded because the inequality is strict. The integers are

$$ 0,\ 1,\ 2,\ 3,\ 4,\ 5,\ 6,\ 7 $$

That is 8 integers. Answer: G

Check With Two Endpoints

• $x = 8$: $|16 - 7| = 9$, and $9 < 9$ is false, so 8 is out.
• $x = 7$: $|14 - 7| = 7 < 9$, true.
• $x = -1$: $|-2 - 7| = 9$, not less than 9, so $-1$ is out.
• $x = 0$: $|0 - 7| = 7 < 9$, true.

Traps on the Answer Choices

• J) 10: counts $-1$ through $8$, which is the answer to $|2x - 7| \le 9$. One symbol changes the answer by 2.
• H) 9: includes one endpoint but not the other.
• F) 7: forgets that $0$ is an integer, or counts $1$ through $7$.
• K) infinitely many: correct for the number of real solutions, but the question asks for integers.

Counting Shortcut

For integers strictly between $a$ and $b$ (both integers), the count is $b - a - 1$. Here $8 - (-1) - 1 = 8$. If both endpoints are included, add 1 back to the difference: $8 - (-1) + 1 = 10$.

Number Line Picture

On a number line, the solution is the open interval centered at $x = 3.5$ (the value where $2x - 7 = 0$) with radius $4.5$ (since $9 / 2 = 4.5$). That gives $3.5 - 4.5 = -1$ and $3.5 + 4.5 = 8$.

Two More to Practice (Same Method)

1. $|3x + 1| \le 11$: $-11 \le 3x + 1 \le 11$, so $-4 \le x \le \dfrac{10}{3}$. The integers are $-4, -3, \dots, 3$, which is 8 integers.
2. $|x - 5| > 2$: $x < 3$ or $x > 7$. A "greater than" absolute value splits into two separate rays.

For a different ACT Math question where one rejected value is the whole point, see the log equation with an extraneous root.