Solve the following inequality algebraically.
5, vertical bar, x, plus, 7, vertical bar, plus, 1, is less than or equal to, 36
5∣x+7∣+1≤36

We are tasked with solving the inequality:

\[
5|x + 7| + 1 \leq 36
\]

### Step-by-step solution:

1. **Isolate the absolute value term:**

   Subtract 1 from both sides:
   \[
   5|x + 7| \leq 35
   \]

2. **Divide by 5:**

   \[
   |x + 7| \leq 7
   \]

3. **Solve the absolute value inequality:**

   Recall that for \( |A| \leq B \), where \( B \geq 0 \), the inequality translates to:
   \[
   -B \leq A \leq B
   \]
   Applying this to \( |x + 7| \leq 7 \), we have:
   \[
   -7 \leq x + 7 \leq 7
   \]

4. **Solve for \( x \):**

   Subtract 7 from all parts of the inequality:
   \[
   -7 - 7 \leq x \leq 7 - 7
   \]
   Simplifying:
   \[
   -14 \leq x \leq 0
   \]

### Final answer:

The solution to the inequality is:

\[
x \in [-14, 0]
\]

Would you like more details, or do you have any further questions?

### 5 Related Questions:
1. How do you solve absolute value inequalities with greater than signs?
2. What happens when there's a negative coefficient in front of the absolute value?
3. Can you graph this inequality on a number line?
4. How do absolute value functions behave differently from regular linear equations?
5. How can this inequality be solved by graphing instead of algebraically?

### Tip:
Always isolate the absolute value term first when solving inequalities that involve absolute values.

Solve the following inequality algebraically: 5|x + 7| + 1 ≤ 36

This step-by-step guide shows how to solve the absolute value inequality 5|x + 7| + 1 ≤ 36 algebraically. Key algebraic concepts include isolating the absolute value term and solving the resulting inequality. Suitable for Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation