To solve the equation \( |4x + 7| = 9 \), we need to consider the definition of the absolute value. The absolute value of an expression represents its distance from zero on the number line, regardless of direction. Therefore, \( |4x + 7| = 9 \) implies that:

1. \( 4x + 7 = 9 \) (the expression inside the absolute value is 9)
2. \( 4x + 7 = -9 \) (the expression inside the absolute value is -9)

Let's solve each case separately.

### Case 1: \( 4x + 7 = 9 \)

\[
4x + 7 = 9
\]

Subtract 7 from both sides:

\[
4x = 2
\]

Divide by 4:

\[
x = \frac{2}{4} = \frac{1}{2}
\]

### Case 2: \( 4x + 7 = -9 \)

\[
4x + 7 = -9
\]

Subtract 7 from both sides:

\[
4x = -16
\]

Divide by 4:

\[
x = \frac{-16}{4} = -4
\]

### Solutions

The two solutions for the equation \( |4x + 7| = 9 \) are:

\[
x = \frac{1}{2} \quad \text{and} \quad x = -4
\]

Would you like more details on these steps, or do you have any other questions?

Here are some related questions you might find interesting:

1. How do you solve equations with absolute values involving variables on both sides?
2. What are the properties of absolute value functions?
3. How do you graph absolute value equations?
4. What is the difference between solving linear equations and absolute value equations?
5. How do absolute value inequalities differ from absolute value equations?

**Tip:** When solving absolute value equations, always consider both the positive and negative scenarios of the expression inside the absolute value.

Learn how to solve the equation |4x + 7| = 9 step-by-step. Understand the concept of absolute value and solve linear equations involving absolute values effectively. Suitable for Grades 9-12 students.

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