Questions: Solve the following inequality and enter your answer using interval notation: 5x+9<9 Answer: x must be in

Solve the following inequality and enter your answer using interval notation: 5x+9<9 Answer: x must be in
Transcript text: Solve the following inequality and enter your answer using interval notation: \[ |5 x+9|<9 \] Answer: $x$ must be in $\square$
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Solution

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Solution Steps

To solve the inequality \(|5x + 9| < 9\), we need to consider the definition of absolute value. This inequality can be split into two separate inequalities: \(5x + 9 < 9\) and \(5x + 9 > -9\). Solving these two inequalities will give us the range of values for \(x\) that satisfy the original inequality. Finally, we express the solution in interval notation.

Step 1: Understand the Inequality

We are given the inequality:

\[ |5x + 9| < 9 \]

This is an absolute value inequality, which means we need to consider two separate cases to solve it.

Step 2: Split the Absolute Value Inequality

The inequality \(|5x + 9| < 9\) can be split into two separate inequalities:

  1. \(5x + 9 < 9\)
  2. \(5x + 9 > -9\)
Step 3: Solve the First Inequality

Let's solve the first inequality:

\[ 5x + 9 < 9 \]

Subtract 9 from both sides:

\[ 5x < 0 \]

Divide both sides by 5:

\[ x < 0 \]

Step 4: Solve the Second Inequality

Now, solve the second inequality:

\[ 5x + 9 > -9 \]

Subtract 9 from both sides:

\[ 5x > -18 \]

Divide both sides by 5:

\[ x > -\frac{18}{5} \]

Step 5: Combine the Solutions

The solution to the inequality \(|5x + 9| < 9\) is the intersection of the solutions to the two inequalities:

\[ -\frac{18}{5} < x < 0 \]

Final Answer

The solution in interval notation is:

\[ \boxed{\left(-\frac{18}{5}, 0\right)} \]

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