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

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.

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.

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

1. \(5x + 9 < 9\)
2. \(5x + 9 > -9\)

Let's solve the first inequality:

\[ 5x + 9 < 9 \]

Subtract 9 from both sides:

\[ 5x < 0 \]

Divide both sides by 5:

\[ x < 0 \]

Now, solve the second inequality:

\[ 5x + 9 > -9 \]

Subtract 9 from both sides:

\[ 5x > -18 \]

Divide both sides by 5:

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

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