To solve the inequality $707 \leq 4|7c - 875| + 7$, let's break it down step by step.

Step 1: Isolate the absolute value expression

Start by subtracting 7 from both sides to isolate the absolute value expression:

$707 - 7 \leq 4|7c - 875|$ $700 \leq 4|7c - 875|$

Now, divide both sides by 4 to further isolate the absolute value term:

$\frac{700}{4} \leq |7c - 875|$ $175 \leq |7c - 875|$

Step 2: Solve the inequality for $c$

The inequality $175 \leq |7c - 875|$ means that the expression inside the absolute value, $7c - 875$, can be either greater than or equal to 175 or less than or equal to -175. Therefore, we need to solve two separate inequalities:

1. $7c - 875 \geq 175$
2. $7c - 875 \leq -175$

Solve inequality 1: $7c - 875 \geq 175$

Add 875 to both sides:

$7c \geq 175 + 875$ $7c \geq 1050$

Now divide both sides by 7:

$c \geq \frac{1050}{7}$ $c \geq 150$

Solve inequality 2: $7c - 875 \leq -175$

Add 875 to both sides:

$7c \leq -175 + 875$ $7c \leq 700$

Now divide both sides by 7:

$c \leq \frac{700}{7}$ $c \leq 100$

Step 3: Combine the results

We have two inequalities:

$c \geq 150 \quad \text{or} \quad c \leq 100$

Thus, the solution to the inequality is:

$c \leq 100 \quad \text{or} \quad c \geq 150$

Graphing the solution:

The solution $c \leq 100$ or $c \geq 150$ can be represented on a number line as two intervals:

• One interval is from negative infinity up to 100 (inclusive).
• The other interval is from 150 to positive infinity (inclusive).

I will now graph the solution on a number line.

Let me know if you'd like a more detailed explanation of any step! And feel free to ask if you have any further questions.

Follow-up questions:

1. How do you interpret inequalities involving absolute values geometrically?
2. Can you solve a similar inequality with multiple absolute values?
3. How would the solution change if the inequality had a different constant, such as 800?
4. What would happen if the inequality used a different operation (e.g., subtraction instead of addition)?
5. Can you solve absolute value equations similarly?

Tip: When solving absolute value inequalities, remember to split them into two cases based on the definition of absolute value.