1. The problem is to understand when to use infinities in solving absolute value equations. 2. Absolute value equations have the form $|x| = a$, where $a \geq 0$. 3. The absolute value $|x|$ represents the distance of $x$ from zero on the number line, so it is always non-negative. 4. To solve $|x| = a$, we split it into two cases: - Case 1: $x = a$ - Case 2: $x = -a$ 5. Infinities are not used directly in solving absolute value equations because the solutions are specific values or intervals, not infinite bounds. 6. However, when solving inequalities involving absolute values, such as $|x| < a$ or $|x| > a$, we use intervals that may extend to infinity: - For $|x| < a$, the solution is $-a < x < a$ (a finite interval). - For $|x| > a$, the solution is $x < -a$ or $x > a$, which involves two intervals extending to $-\infty$ and $+\infty$ respectively. 7. So, infinities are used when solving absolute value inequalities to describe unbounded intervals, but not when solving absolute value equations. Final answer: Use infinities only when solving absolute value inequalities to describe intervals extending indefinitely. For absolute value equations, solutions are finite values without involving infinities.