Question

Solve the inequality
1. |2x - 3| /leq 11

Answer 1

To solve the inequality \( \frac{|2x - 3|}{11} \leq 1 \), we start by multiplying both sides of the inequality by 11 (since 11 is positive, the direction of the inequality does not change):

\[
|2x - 3| \leq 11
\]

Now, we can rewrite the absolute value inequality as a compound inequality:

\[
-11 \leq 2x - 3 \leq 11
\]

Next, we solve the two parts of the compound inequality separately.

**1. For the left part:**

\[
-11 \leq 2x - 3
\]

Adding 3 to both sides:

\[
-11 + 3 \leq 2x \\
-8 \leq 2x
\]

Dividing by 2:

\[
-4 \leq x \quad \text{or} \quad x \geq -4
\]

**2. For the right part:**

\[
2x - 3 \leq 11
\]

Adding 3 to both sides:

\[
2x \leq 11 + 3 \\
2x \leq 14
\]

Dividing by 2:

\[
x \leq 7
\]

**Combining both parts:**

Now, we combine the results:

\[
-4 \leq x \leq 7
\]

So, the solution to the inequality \( \frac{|2x - 3|}{11} \leq 1 \) is:

\[
\boxed{[-4, 7]}
\]