Absolute Value Equation Integer Solutions

1. $x = 0$ is a solution since $2(0)(1) = 5(0)^2 = 0$.
2. If $x \neq 0$, then $|x| > 0$. Divide both sides by $|x|$ (noting $x^2 = |x|^2$): $2(x^2 + 1) = 5|x|$ $2|x|^2 - 5|x| + 2 = 0$ $(2|x| - 1)(|x| - 2) = 0$ So $|x| = \frac{1}{2}$ or $|x| = 2$.

Since $x$ must be an integer:

• $|x| = \frac{1}{2}$ gives no integer solutions.
• $|x| = 2$ gives $x = 2$ and $x = -2$.

Thus, the integer solutions are $x = 0, 2, -2$, making a total of 3 solutions.