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

Transcript text: Solve the inequality. \[ \begin{array}{r} 4 x^{2}-16 \geq 0 \\ x \leq[?] \text { or } x \geq \square \end{array} \]

Solution

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 \(x\).

Step 1: Solve the Equation

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: \[ x = -2 \quad \text{and} \quad x = 2 \]

Step 2: Test Intervals

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

\( (-\infty, -2) \)
\( [-2, 2] \)
\( (2, \infty) \)

Step 3: Determine Where the Inequality Holds

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 \( x = 3 \): \[ 4(3)^2 - 16 = 36 - 16 = 20 \geq 0 \quad \text{(True)} \]

Step 4: Write the Solution

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: \[ (-\infty, -2] \cup [2, \infty) \]