Question

Solve the inequality and graph the solution. |2 x+9| \leq 15

Short answer

The solution to the inequality is -12 \leq x \leq 3.

Step-by-step solution

Write out the two possible inequalities

Because, the definition of an absolute value is \(|a| = a\) if \(a \geq 0\) and \(|a| = -a\) if \(a < 0\), you need to write out two possible inequalities for \(2x+9\). These are \(2x + 9 \leq 15\) and \(2x + 9 \geq -15\).

Solve the first inequality

Solving the first inequality \(2x + 9 \leq 15\) for x. Subtract 9 from both sides to obtain \(2x \leq 6\). Then, divide by the coefficient of x (which is 2) to get \(x \leq 3\).

Solve the second inequality

Solving the second inequality \(2x + 9 \geq -15\) for x. Subtract 9 from both sides to get \(2x \geq -24\). Then, divide by the coefficient of x (which is 2) to get \(x \geq -12\).

Combine the obtained intervals

Combine solutions from step 2 and 3. This gives the solution \( -12 \leq x \leq 3\) for the given inequality.

Graph the solution

On the number line, draw a line segment between -12 and 3 inclusive. This is your solution graph showing all real numbers from -12 to 3 inclusive that are solutions to the inequality.