Solve the Quadratic Inequality: x² - 25 < 0

To solve the inequality $x^2 - 25 < 0$, follow these steps:

• Step 1: Factor the quadratic expression $x^2 - 25$. This is a difference of squares, which factors as $(x - 5)(x + 5)$.
• Step 2: Identify the critical points where the expression equals zero, i.e., where $(x-5)(x+5) = 0$. The solutions are $x = 5$ and $x = -5$.
• Step 3: Determine the intervals defined by these critical points: $(-∞, -5)$, $(-5, 5)$, and $(5, ∞)$.
• Step 4: Test each interval to see where the inequality $(x-5)(x+5) < 0$ holds:
• Choose a test point from $(-∞, -5)$, such as $x = -6$. Then, $(-6-5)(-6+5) = (-11)(-1) > 0$. This interval does not satisfy the inequality.
• Choose a test point from $(-5, 5)$, such as $x = 0$. Then, $(0-5)(0+5) = (-5)(5) = -25 < 0$. This interval satisfies the inequality.
• Choose a test point from $(5, ∞)$, such as $x = 6$. Then, $(6-5)(6+5) = (1)(11) > 0$. This interval does not satisfy the inequality.
• Step 5: Conclude the solution. The inequality holds true in the interval $(-5, 5)$.

Therefore, the solution to the inequality $x^2 - 25 < 0$ is $-5 < x < 5$.