Question

Solve the inequality. $$|2 x-9| \leq 11$$

Short answer

The solution to the inequality \(|2x - 9| \leq 11\) is \(-1 \leq x \leq 10\)

Step-by-step solution

Set up the two cases

Case one: \(2x - 9 \leq 11\) and case two: \(2x - 9 \geq -11\). These are derived from the definition of absolute value, where an absolute value expression is equal to the expression without the absolute value if it’s negative or positive.

Solve the first inequality

Adding 9 to both sides, the inequality in case one becomes \(2x \leq 20\). To get the value of x, divide both sides by 2 to get \(x \leq 10\). This is the solution for case one.

Solve the second inequality

Adding 9 to both sides, the inequality in case two becomes \(2x \geq -2\). Dividing both sides by 2, gives \(x \geq -1\). This is the solution for case two.

Combine the solutions

Since the original inequality is a 'less than or equal to' inequality, this means we're interested in the overlap of the two cases' solutions. The solution for the original inequality, then, will be the intersection of the solutions from case one and case two, which is \(-1 \leq x \leq 10\)