To solve the absolute inequality $\frac{4x - 7}{4} \leq \frac{1}{3}$, follow these steps:

Step 1: Eliminate the denominator

To eliminate the denominator of 4 on the left-hand side, multiply both sides of the inequality by 4:

$4 \times \frac{4x - 7}{4} \leq 4 \times \frac{1}{3}$

This simplifies to:

$4x - 7 \leq \frac{4}{3}$

Step 2: Isolate the variable

Now, add 7 to both sides of the inequality to move the constant to the other side:

$4x \leq \frac{4}{3} + 7$

We need to convert 7 into a fraction with denominator 3:

$7 = \frac{21}{3}$

Now add the fractions:

$4x \leq \frac{4}{3} + \frac{21}{3} = \frac{25}{3}$

Step 3: Solve for $x$

Next, divide both sides by 4 to isolate $x$:

$x \leq \frac{\frac{25}{3}}{4} = \frac{25}{12}$

Thus, the solution is:

$x \leq \frac{25}{12}$

Final Answer:

The solution to the inequality is $x \leq \frac{25}{12}$.

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Would you like more details or further explanation on any step?

Here are 5 related questions:

1. How do you solve absolute inequalities involving two terms?
2. What happens if the inequality has a negative sign?
3. Can absolute inequalities have multiple solutions?
4. How do you convert improper fractions to mixed numbers?
5. How do you graph an inequality on a number line?

Tip: Always remember to reverse the inequality when multiplying or dividing both sides by a negative number.