Solve the following absolute value inequality.
StartAbsoluteValue 14 x EndAbsoluteValueless than or equals7

The given absolute value inequality is:

\[
|14x| \leq 7
\]

To solve this, we need to consider the definition of absolute value. The inequality \( |A| \leq B \) can be rewritten as:

\[
-B \leq A \leq B
\]

In this case, \( A = 14x \) and \( B = 7 \), so we have:

\[
-7 \leq 14x \leq 7
\]

Next, we solve for \( x \) by dividing all parts of the inequality by 14:

\[
\frac{-7}{14} \leq \frac{14x}{14} \leq \frac{7}{14}
\]

Simplifying this, we get:

\[
-\frac{1}{2} \leq x \leq \frac{1}{2}
\]

So, the solution to the inequality is:

\[
x \in \left[ -\frac{1}{2}, \frac{1}{2} \right]
\]

This means that \( x \) can be any value between \(-\frac{1}{2}\) and \(\frac{1}{2}\), inclusive.

Would you like any further details or have any questions?

Here are some related questions:

1. How do you solve absolute value inequalities involving "greater than"?
2. What is the difference between solving \( |x| \leq a \) and \( |x| \geq a \)?
3. How do you graph the solution to an absolute value inequality on a number line?
4. Can absolute value inequalities have no solution? If so, when?
5. How would the inequality change if it was \( |14x| > 7 \) instead?

**Tip:** When solving absolute value inequalities, remember that you often need to split the problem into two cases to remove the absolute value.

Solve the following absolute value inequality: |14x| ≤ 7

Understand how to solve the absolute value inequality |14x| ≤ 7 with a clear, step-by-step solution. This problem explores key algebraic concepts such as absolute value inequalities and their simplification. Suitable for students in Grades 8-10.

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