Use the given functions to find all values of \( x \) that satisfy the inequality \( f(x) \geq g(x) \). \[ f(x)=5 x^{2} \] \( g(x)=22 x-8 \)

To solve the inequality \( f(x) \geq g(x) \), we first set up the equation: \[ 5x^2 \geq 22x - 8. \] Rearranging this gives us: \[ 5x^2 - 22x + 8 \geq 0. \] Next, we will find the roots of the equation \( 5x^2 - 22x + 8 = 0 \) using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \): Here, \( a = 5 \), \( b = -22 \), and \( c = 8 \). Calculating the discriminant: \[ b^2 - 4ac = (-22)^2 - 4 \cdot 5 \cdot 8 = 484 - 160 = 324. \] Now, applying the quadratic formula: \[ x = \frac{22 \pm \sqrt{324}}{2 \cdot 5} = \frac{22 \pm 18}{10}. \] This gives us two solutions: \[ x_1 = \frac{40}{10} = 4 \quad \text{and} \quad x_2 = \frac{4}{10} = 0.4. \] Now, we will test intervals determined by these roots: \( (-\infty, 0.4) \), \( (0.4, 4) \), and \( (4, \infty) \). 1. For \( x < 0.4 \), let’s test \( x = 0 \): \[ 5(0)^2 - 22(0) + 8 = 8 \geq 0 \quad \text{(True)} \] 2. For \( 0.4 < x < 4 \), let’s test \( x = 1 \): \[ 5(1)^2 - 22(1) + 8 = 5 - 22 + 8 = -9 \geq 0 \quad \text{(False)} \] 3. For \( x > 4 \), let’s test \( x = 5 \): \[ 5(5)^2 - 22(5) + 8 = 125 - 110 + 8 = 23 \geq 0 \quad \text{(True)} \] From our test, we find the inequality holds for \( x \) in the intervals \( (-\infty, 0.4] \) and \( [4, \infty) \). Thus, the solution to the inequality \( f(x) \geq g(x) \) is: \[ x \in (-\infty, 0.4] \cup [4, \infty). \]