To solve the inequality $|5 - 7x| \leq 12$, we’ll break it down into two cases due to the absolute value.

The inequality $|5 - 7x| \leq 12$ means: $-12 \leq 5 - 7x \leq 12$

Now, we solve these two inequalities separately.

Step 1: Solve $-12 \leq 5 - 7x$

1. Subtract 5 from both sides: $-17 \leq -7x$
2. Divide by $-7$ and reverse the inequality sign: $x \leq \frac{17}{7}$

Step 2: Solve $5 - 7x \leq 12$

1. Subtract 5 from both sides: $-7x \leq 7$
2. Divide by $-7$ and reverse the inequality sign: $x \geq -1$

Combine the Solutions

The solution to $-12 \leq 5 - 7x \leq 12$ is: $-1 \leq x \leq \frac{17}{7}$

Interval Notation

The solution in interval notation is: $\left[-1, \frac{17}{7}\right]$

Answer

The correct answer is:

• A. The solution set is $\left[-1, \frac{17}{7}\right]$.

Would you like further details or explanations?

Here are five related questions to explore further:

1. How would the solution change if the inequality were $|5 - 7x| < 12$?
2. How do you solve absolute value inequalities when the inequality sign is "greater than" instead of "less than"?
3. What happens to the solution if the inequality becomes $|5 - 7x| \geq 12$?
4. How do you interpret the solution interval $\left[-1, \frac{17}{7}\right]$ on a number line?
5. Can you explain why we reverse the inequality when dividing by a negative number?

Tip: Always remember to flip the inequality sign when dividing or multiplying both sides by a negative value in inequalities.