Questions: Solve. Write your answer in interval notation. 3x+5>8 If no solutions exists - enter No solutions. Use reduced fractions in your answer- no decimals!

Solve. Write your answer in interval notation. 3x+5>8 If no solutions exists - enter No solutions. Use reduced fractions in your answer- no decimals!
Transcript text: Solve. Write your answer in interval notation. \[ |3 x+5|>8 \] $\square$ If no solutions exists - enter No solutions. Use reduced fractions in your answer- no decimals!
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Solution

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Solution Steps

To solve the inequality \(|3x + 5| > 8\), we need to consider two separate cases due to the absolute value. The first case is when \(3x + 5 > 8\) and the second case is when \(3x + 5 < -8\). We will solve each inequality separately and then combine the solutions. Finally, we will express the solution in interval notation.

Step 1: Understand the Absolute Value Inequality

The given inequality is:

\[ |3x + 5| > 8 \]

An absolute value inequality of the form \(|A| > B\) can be split into two separate inequalities:

  1. \(A > B\)
  2. \(A < -B\)
Step 2: Split the Inequality

For the inequality \(|3x + 5| > 8\), we split it into two separate inequalities:

  1. \(3x + 5 > 8\)
  2. \(3x + 5 < -8\)
Step 3: Solve the First Inequality

Solve \(3x + 5 > 8\):

\[ 3x + 5 > 8 \]

Subtract 5 from both sides:

\[ 3x > 3 \]

Divide both sides by 3:

\[ x > 1 \]

Step 4: Solve the Second Inequality

Solve \(3x + 5 < -8\):

\[ 3x + 5 < -8 \]

Subtract 5 from both sides:

\[ 3x < -13 \]

Divide both sides by 3:

\[ x < -\frac{13}{3} \]

Step 5: Combine the Solutions

The solutions to the inequality \(|3x + 5| > 8\) are the union of the solutions to the two inequalities:

  1. \(x > 1\)
  2. \(x < -\frac{13}{3}\)

In interval notation, this is:

\[ (-\infty, -\frac{13}{3}) \cup (1, \infty) \]

Final Answer

The solution in interval notation is:

\[ \boxed{(-\infty, -\frac{13}{3}) \cup (1, \infty)} \]

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