Question - Solving Absolute Value Inequality
Solution:
To solve the inequality $$ 3 \left| \frac{x+3}{2} \right| \leq 9 $$, we can proceed as follows:1. Divide both sides of the inequality by 3 to isolate the absolute value expression:\[ \left| \frac{x+3}{2} \right| \leq \frac{9}{3} \]\[ \left| \frac{x+3}{2} \right| \leq 3 \]2. The absolute value inequality $$ |A| \leq B $$ where $$ B $$ is non-negative (B ≥ 0) can be split into two inequalities:\[ -B \leq A \leq B \]Applying this concept, we get:\[ -3 \leq \frac{x+3}{2} \leq 3 \]3. Multiply all parts of the compound inequality by 2 to get rid of the fraction:\[ -6 \leq x+3 \leq 6 \]4. Now subtract 3 from all parts of the compound inequality to solve for $$ x $$:\[ -6 - 3 \leq x+3 - 3 \leq 6 - 3 \]\[ -9 \leq x \leq 3 \]So the solution set for the inequality is $$ x \in [-9, 3] $$.
CamTutor
In regards to math, we are professionals.
Get In Touch
Email: camtutor.ai@gmail.com