Two Step Inequalities Worksheets - Free Printable

To solve the problem, we need to evaluate each inequality and determine which of the given numbers satisfy it. Let's go through each inequality step by step.

---

1. \( \frac{3}{4}x < 6 \)

#### Solve for \( x \):
\[
\frac{3}{4}x < 6
\]
Multiply both sides by \( \frac{4}{3} \) to isolate \( x \):
\[
x < 6 \cdot \frac{4}{3}
\]
\[
x < 8
\]

#### Possible values:
The possible values are those less than 8. From the given options: \( 5, 7, 9, 11 \), the values that satisfy \( x < 8 \) are:
\[
\boxed{5, 7}
\]

---

2. \( 10 - 3x \leq -13 \)

#### Solve for \( x \):
\[
10 - 3x \leq -13
\]
Subtract 10 from both sides:
\[
-3x \leq -13 - 10
\]
\[
-3x \leq -23
\]
Divide both sides by \(-3\) (and reverse the inequality sign):
\[
x \geq \frac{23}{3}
\]
\[
x \geq 7.67
\]

#### Possible values:
The possible values are those greater than or equal to 7.67. From the given options: \( 1, 3, 5, 7 \), none of these values satisfy \( x \geq 7.67 \). Therefore, there are no solutions in this case:
\[
\boxed{\text{None}}
\]

---

3. \( 7 > 3x + 8 \)

#### Solve for \( x \):
\[
7 > 3x + 8
\]
Subtract 8 from both sides:
\[
7 - 8 > 3x
\]
\[
-1 > 3x
\]
Divide both sides by 3:
\[
-\frac{1}{3} > x
\]
\[
x < -\frac{1}{3}
\]

#### Possible values:
The possible values are those less than \(-\frac{1}{3}\). From the given options: \( -2, -1, 0, 1 \), the values that satisfy \( x < -\frac{1}{3} \) are:
\[
\boxed{-2, -1}
\]

---

4. \( 2x + 1 \leq 15 \)

#### Solve for \( x \):
\[
2x + 1 \leq 15
\]
Subtract 1 from both sides:
\[
2x \leq 14
\]
Divide both sides by 2:
\[
x \leq 7
\]

#### Possible values:
The possible values are those less than or equal to 7. From the given options: \( 3, 5, 7, 9 \), the values that satisfy \( x \leq 7 \) are:
\[
\boxed{3, 5, 7}
\]

---

5. \( x - 13 > -5 \)

#### Solve for \( x \):
\[
x - 13 > -5
\]
Add 13 to both sides:
\[
x > -5 + 13
\]
\[
x > 8
\]

#### Possible values:
The possible values are those greater than 8. From the given options: \( -2, -1, 0, 1 \), none of these values satisfy \( x > 8 \). Therefore, there are no solutions in this case:
\[
\boxed{\text{None}}
\]

---

6. \( \frac{3}{5}x \geq 3 \)

#### Solve for \( x \):
\[
\frac{3}{5}x \geq 3
\]
Multiply both sides by \( \frac{5}{3} \):
\[
x \geq 3 \cdot \frac{5}{3}
\]
\[
x \geq 5
\]

#### Possible values:
The possible values are those greater than or equal to 5. From the given options: \( -3, -1, 3, 5 \), the values that satisfy \( x \geq 5 \) are:
\[
\boxed{5}
\]

---

7. \( 12 + 2x \leq 18 \)

#### Solve for \( x \):
\[
12 + 2x \leq 18
\]
Subtract 12 from both sides:
\[
2x \leq 18 - 12
\]
\[
2x \leq 6
\]
Divide both sides by 2:
\[
x \leq 3
\]

#### Possible values:
The possible values are those less than or equal to 3. From the given options: \( 4, 6, 8, 10 \), none of these values satisfy \( x \leq 3 \). Therefore, there are no solutions in this case:
\[
\boxed{\text{None}}
\]

---

8. \( 2 - x \geq -5 \)

#### Solve for \( x \):
\[
2 - x \geq -5
\]
Subtract 2 from both sides:
\[
-x \geq -5 - 2
\]
\[
-x \geq -7
\]
Multiply both sides by \(-1\) (and reverse the inequality sign):
\[
x \leq 7
\]

#### Possible values:
The possible values are those less than or equal to 7. From the given options: \( -4, -2, 0, 2 \), all of these values satisfy \( x \leq 7 \):
\[
\boxed{-4, -2, 0, 2}
\]

---

9. \( 5 = 5x + 2 \)

#### Solve for \( x \):
\[
5 = 5x + 2
\]
Subtract 2 from both sides:
\[
5 - 2 = 5x
\]
\[
3 = 5x
\]
Divide both sides by 5:
\[
x = \frac{3}{5}
\]

#### Possible values:
The only value that satisfies \( x = \frac{3}{5} \) is \( \frac{3}{5} \). From the given options: \( 1, 3, 5, 7 \), none of these values satisfy \( x = \frac{3}{5} \). Therefore, there are no solutions in this case:
\[
\boxed{\text{None}}
\]

---

10. \( 2x + 7 \geq 21 \)

#### Solve for \( x \):
\[
2x + 7 \geq 21
\]
Subtract 7 from both sides:
\[
2x \geq 21 - 7
\]
\[
2x \geq 14
\]
Divide both sides by 2:
\[
x \geq 7
\]

#### Possible values:
The possible values are those greater than or equal to 7. From the given options: \( 1, 7, 9, 11 \), the values that satisfy \( x \geq 7 \) are:
\[
\boxed{7, 9, 11}
\]

---

11. \( 6 + 5x \leq 11 \)

#### Solve for \( x \):
\[
6 + 5x \leq 11
\]
Subtract 6 from both sides:
\[
5x \leq 11 - 6
\]
\[
5x \leq 5
\]
Divide both sides by 5:
\[
x \leq 1
\]

#### Possible values:
The possible values are those less than or equal to 1. From the given options: \( 3, 1, -1, -3 \), the values that satisfy \( x \leq 1 \) are:
\[
\boxed{1, -1, -3}
\]

---

12. \( 3x - 9 > 21 \)

#### Solve for \( x \):
\[
3x - 9 > 21
\]
Add 9 to both sides:
\[
3x > 21 + 9
\]
\[
3x > 30
\]
Divide both sides by 3:
\[
x > 10
\]

#### Possible values:
The possible values are those greater than 10. From the given options: \( 16, 18, 20, 22 \), all of these values satisfy \( x > 10 \):
\[
\boxed{16, 18, 20, 22}
\]

---

13. \( 15 \geq 5x - 7 \)

#### Solve for \( x \):
\[
15 \geq 5x - 7
\]
Add 7 to both sides:
\[
15 + 7 \geq 5x
\]
\[
22 \geq 5x
\]
Divide both sides by 5:
\[
\frac{22}{5} \geq x
\]
\[
x \leq 4.4
\]

#### Possible values:
The possible values are those less than or equal to 4.4. From the given options: \( 1, 3, 5, 7 \), the values that satisfy \( x \leq 4.4 \) are:
\[
\boxed{1, 3}
\]

---

14. \( 4x - 2 \leq 10 \)

#### Solve for \( x \):
\[
4x - 2 \leq 10
\]
Add 2 to both sides:
\[
4x \leq 10 + 2
\]
\[
4x \leq 12
\]
Divide both sides by 4:
\[
x \leq 3
\]

#### Possible values:
The possible values are those less than or equal to 3. From the given options: \( 1, 2, 3, 4 \), the values that satisfy \( x \leq 3 \) are:
\[
\boxed{1, 2, 3}
\]

---

Final Answer:

\[
\boxed{
\begin{array}{l}
1. 5, 7 \\
2. \text{None} \\
3. -2, -1 \\
4. 3, 5, 7 \\
5. \text{None} \\
6. 5 \\
7. \text{None} \\
8. -4, -2, 0, 2 \\
9. \text{None} \\
10. 7, 9, 11 \\
11. 1, -1, -3 \\
12. 16, 18, 20, 22 \\
13. 1, 3 \\
14. 1, 2, 3 \\
\end{array}
}
\]