Solve for \( x \) and illustrate your answer on a number line. \( 5.1 \quad x^{2} \leq 64 \) \( 5.2 \quad 81-4 x^{2} \geq 0 \) \( 5.3 \quad x^{2}-x-12>0 \) \( 5.4 \quad x^{2}-7 x \geq 0 \)

Sure! Let's solve each of the inequalities step by step and illustrate their solutions on a number line. --- ### **5.1. \( x^{2} \leq 64 \)** **Solution:** 1. **Solve the inequality:** \[ x^{2} \leq 64 \] Take the square root of both sides: \[ |x| \leq \sqrt{64} \\ |x| \leq 8 \] This implies: \[ -8 \leq x \leq 8 \] **Number Line Illustration:** ``` <----|========|========|========|========|----------> -8 0 8 ``` - The shaded region between -8 and 8 represents all real numbers \( x \) such that \( -8 \leq x \leq 8 \). - Solid circles at -8 and 8 indicate that these endpoints are included in the solution (since the inequality is ≤). --- ### **5.2. \( 81 - 4x^{2} \geq 0 \)** **Solution:** 1. **Rearrange the inequality:** \[ 81 - 4x^{2} \geq 0 \\ -4x^{2} \geq -81 \\ \] Multiply both sides by -1 (note that the inequality sign reverses when multiplying by a negative number): \[ 4x^{2} \leq 81 \\ x^{2} \leq \frac{81}{4} \\ x^{2} \leq 20.25 \] 2. **Take the square root of both sides:** \[ |x| \leq \sqrt{20.25} \\ |x| \leq 4.5 \] This implies: \[ -4.5 \leq x \leq 4.5 \] **Number Line Illustration:** ``` <----|=====|=====|=====|=====|-----> -4.5 0 4.5 ``` - The shaded region between -4.5 and 4.5 represents all real numbers \( x \) such that \( -4.5 \leq x \leq 4.5 \). - Solid circles at -4.5 and 4.5 indicate that these endpoints are included in the solution. --- ### **5.3. \( x^{2} - x - 12 > 0 \)** **Solution:** 1. **Factor the quadratic inequality:** \[ x^{2} - x - 12 > 0 \] Factor: \[ (x - 4)(x + 3) > 0 \] 2. **Find the critical points:** \[ x - 4 = 0 \Rightarrow x = 4 \\ x + 3 = 0 \Rightarrow x = -3 \] 3. **Determine the intervals to test:** The critical points divide the number line into three intervals: - \( x < -3 \) - \( -3 < x < 4 \) - \( x > 4 \) 4. **Test each interval:** - **Interval 1: \( x < -3 \)** (e.g., \( x = -4 \)) \[ (-4 - 4)(-4 + 3) = (-8)(-1) = 8 > 0 \quad \text{(True)} \] - **Interval 2: \( -3 < x < 4 \)** (e.g., \( x = 0 \)) \[ (0 - 4)(0 + 3) = (-4)(3) = -12 < 0 \quad \text{(False)} \] - **Interval 3: \( x > 4 \)** (e.g., \( x = 5 \)) \[ (5 - 4)(5 + 3) = (1)(8) = 8 > 0 \quad \text{(True)} \] 5. **Solution:** The inequality \( (x - 4)(x + 3) > 0 \) is satisfied in: \[ x < -3 \quad \text{or} \quad x > 4 \] **Number Line Illustration:** ``` <====| |==========| -3 4 ``` - Shaded regions extend to the left of -3 and to the right of 4. - Open circles at -3 and 4 indicate that these points are **not** included in the solution (since the inequality is >, not ≥). --- ### **5.4. \( x^{2} - 7x \geq 0 \)** **Solution:** 1. **Factor the quadratic inequality:** \[ x^{2} - 7x \geq 0 \\ x(x - 7) \geq 0 \] 2. **Find the critical points:** \[ x = 0 \\ x - 7 = 0 \Rightarrow x = 7 \] 3. **Determine the intervals to test:** The critical points divide the number line into three intervals: - \( x < 0 \) - \( 0 < x < 7 \) - \( x > 7 \) 4. **Test each interval:** - **Interval 1: \( x < 0 \)** (e.g., \( x = -1 \)) \[ (-1)(-1 - 7) = (-1)(-8) = 8 \geq 0 \quad \text{(True)} \] - **Interval 2: \( 0 < x < 7 \)** (e.g., \( x = 3 \)) \[ (3)(3 - 7) = (3)(-4) = -12 \geq 0 \quad \text{(False)} \] - **Interval 3: \( x > 7 \)** (e.g., \( x = 8 \)) \[ (8)(8 - 7) = (8)(1) = 8 \geq 0 \quad \text{(True)} \] 5. **Include critical points if they satisfy the inequality:** - At \( x = 0 \): \[ 0(0 - 7) = 0 \geq 0 \quad \text{(True)} \] - At \( x = 7 \): \[ 7(7 - 7) = 7(0) = 0 \geq 0 \quad \text{(True)} \] 6. **Solution:** The inequality \( x(x - 7) \geq 0 \) is satisfied in: \[ x \leq 0 \quad \text{or} \quad x \geq 7 \] **Number Line Illustration:** ``` <====|==========| 0 7 ``` - Shaded regions extend to the left of 0 and to the right of 7. - Solid circles at 0 and 7 indicate that these points are included in the solution (since the inequality is ≥). --- ### **Summary of Solutions:** 1. **5.1:** \( -8 \leq x \leq 8 \) 2. **5.2:** \( -4.5 \leq x \leq 4.5 \) 3. **5.3:** \( x < -3 \) or \( x > 4 \) 4. **5.4:** \( x \leq 0 \) or \( x \geq 7 \) These solutions are visually represented on their respective number lines, illustrating the ranges of \( x \) that satisfy each inequality.