Answer
Here are the solutions to the inequalities:
1. **5.1. \( x^{2} \leq 64 \):**
\( -8 \leq x \leq 8 \)
2. **5.2. \( 81 - 4x^{2} \geq 0 \):**
\( -4.5 \leq x \leq 4.5 \)
3. **5.3. \( x^{2} - x - 12 > 0 \):**
\( x < -3 \) or \( x > 4 \)
4. **5.4. \( x^{2} - 7x \geq 0 \):**
\( x \leq 0 \) or \( x \geq 7 \)
These solutions are illustrated on number lines showing the ranges of \( x \) that satisfy each inequality.
Solution
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.
Reviewed and approved by the UpStudy tutoring team
Explain
Simplify this solution
