Questions: Solve the inequality. 4 x^2 - 16 ≥ 0 x ≤ [?] or x ≥ □

Solution Steps

To solve the inequality \(4x^2 - 16 \geq 0\), we first need to find the values of \(x\) where the expression equals zero. This involves solving the equation \(4x^2 - 16 = 0\). Once we have the critical points, we can test intervals around these points to determine where the inequality holds true. The solution will be in the form of intervals for \(x\).

To find the critical points, we solve the equation: \[ 4x^2 - 16 = 0 \] Factoring gives: \[ 4(x^2 - 4) = 0 \implies x^2 - 4 = 0 \implies (x - 2)(x + 2) = 0 \] Thus, the critical points are: \[ x = -2 \quad \text{and} \quad x = 2 \]

We need to test the intervals determined by the critical points:

1. \( (-\infty, -2) \)
2. \( [-2, 2] \)
3. \( (2, \infty) \)

• For \( x < -2 \): Choose \( x = -3 \): \[ 4(-3)^2 - 16 = 36 - 16 = 20 \geq 0 \quad \text{(True)} \]
• For \( -2 \leq x \leq 2 \): Choose \( x = 0 \): \[ 4(0)^2 - 16 = -16 \geq 0 \quad \text{(False)} \]
• For \( x > 2 \): Choose \( x = 3 \): \[ 4(3)^2 - 16 = 36 - 16 = 20 \geq 0 \quad \text{(True)} \]

The solution to the inequality \(4x^2 - 16 \geq 0\) is: \[ x \leq -2 \quad \text{or} \quad x \geq 2 \] In interval notation, this is expressed as: \[ (-\infty, -2] \cup [2, \infty) \]

Final Answer