Question

Solve the inequalitie, |(3x-1)/4| greater than or equal to 3

Answer 1

To solve the inequality

\[
\left| \frac{3x - 1}{4} \right| \geq 3,
\]

we start by considering the definition of absolute value. The expression inside the absolute value can be either greater than or equal to 3 or less than or equal to -3. This gives us two cases to consider:

**Case 1:**

\[
\frac{3x - 1}{4} \geq 3
\]

To eliminate the fraction, multiply both sides by 4:

\[
3x - 1 \geq 12.
\]

Now, add 1 to both sides:

\[
3x \geq 13.
\]

Next, divide by 3:

\[
x \geq \frac{13}{3}.
\]

**Case 2:**

\[
\frac{3x - 1}{4} \leq -3.
\]

Again, multiply both sides by 4:

\[
3x - 1 \leq -12.
\]

Add 1 to both sides:

\[
3x \leq -11.
\]

Now, divide by 3:

\[
x \leq -\frac{11}{3}.
\]

Now we combine the results from both cases. The final solution for the inequality

\[
\left| \frac{3x - 1}{4} \right| \geq 3
\]

is:

\[
x \leq -\frac{11}{3} \quad \text{or} \quad x \geq \frac{13}{3}.
\]

In interval notation, the solution is:

\[
(-\infty, -\frac{11}{3}] \cup [\frac{13}{3}, \infty).
\]