Question

Worksheet: Solving Inequalities

Solve for x; graph the solution and give the interval for each inequality

1) 5x+6>16 2) 4x – 2 10 3) -4x-9<19


4) -6x + 5 -7 5) 7x-3>-24 6) -12x-6 -66


7) 6+2x 12+8x-3x 8) -x2 + 5 10 9) x3-7<-6


10) 5+4x x+8 11) -30+5x>4(6+8x) 12) -x4-5<17


13) -7x-8 6(1-2x)+1 14) 4(8-2x)-2x 32


15) 5x-7(x+1)> -9 16) -27+8x< -5(1-6x)

Answer 1

Let's solve each inequality step by step and provide the graphing intervals.

### 1) \( 5x + 6 > 16 \)

- Subtract 6 from both sides:
\[
5x > 10
\]
- Divide both sides by 5:
\[
x > 2
\]
- Interval: \( (2, \infty) \)

### 2) \( 4x - 2 < 10 \)

- Add 2 to both sides:
\[
4x < 12
\]
- Divide by 4:
\[
x < 3
\]
- Interval: \( (-\infty, 3) \)

### 3) \( -4x - 9 < 19 \)

- Add 9 to both sides:
\[
-4x < 28
\]
- Divide by -4 (remember to flip the inequality):
\[
x > -7
\]
- Interval: \( (-7, \infty) \)

### 4) \( -6x + 5 - 7 < 0 \)

- Simplify:
\[
-6x - 2 < 0
\]
- Add 2:
\[
-6x < 2
\]
- Divide by -6 (flip the inequality):
\[
x > -\frac{1}{3}
\]
- Interval: \( \left(-\frac{1}{3}, \infty\right) \)

### 5) \( 7x - 3 > -24 \)

- Add 3:
\[
7x > -21
\]
- Divide by 7:
\[
x > -3
\]
- Interval: \( (-3, \infty) \)

### 6) \( -12x - 6 < -66 \)

- Add 6:
\[
-12x < -60
\]
- Divide by -12 (flip the inequality):
\[
x > 5
\]
- Interval: \( (5, \infty) \)

### 7) \( 6 + 2x < 12 + 8x - 3x \)

- Combine like terms:
\[
6 + 2x < 12 + 5x
\]
- Subtract \( 2x \) from both sides:
\[
6 < 12 + 3x
\]
- Subtract 12:
\[
-6 < 3x
\]
- Divide by 3:
\[
x > -2
\]
- Interval: \( (-2, \infty) \)

### 8) \( -x^2 + 5 < 10 \)

- Subtract 5:
\[
-x^2 < 5
\]
- Multiply by -1 (flip the inequality):
\[
x^2 > -5
\]
- This inequality is always true since \( x^2 \) is always non-negative.
- Interval: \( (-\infty, \infty) \)

### 9) \( x^3 - 7 < -6 \)

- Add 7:
\[
x^3 < 1
\]
- Taking the cube root:
\[
x < 1
\]
- Interval: \( (-\infty, 1) \)

### 10) \( 5 + 4x < x + 8 \)

- Subtract \( x \) from both sides:
\[
5 + 3x < 8
\]
- Subtract 5:
\[
3x < 3
\]
- Divide by 3:
\[
x < 1
\]
- Interval: \( (-\infty, 1) \)

### 11) \( -30 + 5x > 4(6 + 8x) \)

- Expand:
\[
-30 + 5x > 24 + 32x
\]
- Subtract \( 5x \) from both sides:
\[
-30 > 24 + 27x
\]
- Subtract 24:
\[
-54 > 27x
\]
- Divide by 27 (flip the inequality):
\[
x < -2
\]
- Interval: \( (-\infty, -2) \)

### 12) \( -x^4 - 5 < 17 \)

- Add 5:
\[
-x^4 < 22
\]
- Multiply by -1 (flip the inequality):
\[
x^4 > -22
\]
- This inequality is always true since \( x^4 \) is always non-negative.
- Interval: \( (-\infty, \infty) \)

### 13) \( -7x - 8 < 6(1 - 2x) + 1 \)

- Expand the right side:
\[
-7x - 8 < 6 - 12x + 1
\]
Simplify:
\[
-7x - 8 < 7 - 12x
\]
- Add \( 12x \) to both sides:
\[
5x - 8 < 7
\]
- Add 8:
\[
5x < 15
\]
- Divide by 5:
\[
x < 3
\]
- Interval: \( (-\infty, 3) \)

### 14) \( 4(8 - 2x) - 2x < 32 \)

- Expand:
\[
32 - 8x - 2x < 32
\]
- Combine:
\[
32 - 10x < 32
\]
- Subtract 32:
\[
-10x < 0
\]
- Divide by -10 (flip the inequality):
\[
x > 0
\]
- Interval: \( (0, \infty) \)

### 15) \( 5x - 7(x + 1) > -9 \)

- Expand:
\[
5x - 7x - 7 > -9
\]
- Combine:
\[
-2x - 7 > -9
\]
- Add 7:
\[
-2x > -2
\]
- Divide by -2 (flip the inequality):
\[
x < 1
\]
- Interval: \( (-\infty, 1) \)

### 16) \( -27 + 8x < -5(1 - 6x) \)

- Expand:
\[
-27 + 8x < -5 + 30x
\]
- Subtract 8x:
\[
-27 < -5 + 22x
\]
- Add 5:
\[
-22 < 22x
\]
- Divide by 22:
\[
-1 < x \quad \text{or} \quad x > -1
\]
- Interval: \( (-1, \infty) \)

### Summary of Solutions and Intervals:

1. \( x > 2 \) \(\Rightarrow (2, \infty)\)
2. \( x < 3 \) \(\Rightarrow (-\infty, 3)\)
3. \( x > -7 \) \(\Rightarrow (-7, \infty)\)
4. \( x > -\frac{1}{3} \) \(\Rightarrow \left(-\frac{1}{3}, \infty\right)\)
5. \( x > -3 \) \(\Rightarrow (-3, \infty)\)
6. \( x > 5 \) \(\Rightarrow (5, \infty)\)
7. \( x > -2 \) \(\Rightarrow (-2, \infty)\)
8. Always true \(\Rightarrow (-\infty, \infty)\)
9. \( x < 1 \) \(\Rightarrow (-\infty, 1)\)
10. \( x < 1 \) \(\Rightarrow (-\infty, 1)\)
11. \( x < -2 \) \(\Rightarrow (-\infty, -2)\)
12. Always true \(\Rightarrow (-\infty, \infty)\)
13. \( x < 3 \) \(\Rightarrow (-\infty, 3)\)
14. \( x > 0 \) \(\Rightarrow (0, \infty)\)
15. \( x < 1 \) \(\Rightarrow (-\infty, 1)\)
16. \( x > -1 \) \(\Rightarrow (-1, \infty)\)

Feel free to graph these intervals on a number line as further practice!