Solve the Quadratic Inequality: x²-9<0 Step by Step

To solve the inequality $x^2 - 9 < 0$, we will perform the following steps:

• Step 1: Factor the inequality $x^2 - 9 = (x - 3)(x + 3)$.
• Step 2: Identify the critical values from the factored expression, which occur at $x = 3$ and $x = -3$.
• Step 3: Use these critical points to divide the number line into intervals: $(-\infty, -3)$, $(-3, 3)$, and $(3, \infty)$.
• Step 4: Test each interval to determine where the inequality holds:
  • For $x = 0$ in the interval $(-3, 3)$, $(0 - 3)(0 + 3) = -9$, which satisfies $< 0$.
  • For $x = -4$ in $(-\infty, -3)$, $(-4 - 3)(-4 + 3) = 7$, which does not satisfy $< 0$.
  • For $x = 4$ in $(3, \infty)$, $(4 - 3)(4 + 3) = 7$, which does not satisfy $< 0$.

Therefore, the inequality $x^2 - 9 < 0$ holds in the interval $-3 < x < 3$. This means any $x$ that falls between these values will satisfy the inequality.

The correct answer is $\mathbf{-3 < x < 3}$.