Solving Equations & Inequalities - Free Printable

To solve equations with unknowns on both sides, the general approach is to isolate the variable (usually \( x \)) on one side of the equation. Here's a step-by-step explanation and solution for each equation:

---

Equation 1: \( 7x - 3 = 3x + 8 \)

1. Subtract \( 3x \) from both sides:
\[
7x - 3x - 3 = 3x - 3x + 8
\]
\[
4x - 3 = 8
\]

2. Add 3 to both sides:
\[
4x - 3 + 3 = 8 + 3
\]
\[
4x = 11
\]

3. Divide by 4:
\[
x = \frac{11}{4}
\]

Solution: \( x = \frac{11}{4} \)

---

Equation 2: \( 5x + 4 = 2x + 9 \)

1. Subtract \( 2x \) from both sides:
\[
5x - 2x + 4 = 2x - 2x + 9
\]
\[
3x + 4 = 9
\]

2. Subtract 4 from both sides:
\[
3x + 4 - 4 = 9 - 4
\]
\[
3x = 5
\]

3. Divide by 3:
\[
x = \frac{5}{3}
\]

Solution: \( x = \frac{5}{3} \)

---

Equation 3: \( 6x - 2 = x + 8 \)

1. Subtract \( x \) from both sides:
\[
6x - x - 2 = x - x + 8
\]
\[
5x - 2 = 8
\]

2. Add 2 to both sides:
\[
5x - 2 + 2 = 8 + 2
\]
\[
5x = 10
\]

3. Divide by 5:
\[
x = 2
\]

Solution: \( x = 2 \)

---

Equation 4: \( 8x + 1 = 3x + 2 \)

1. Subtract \( 3x \) from both sides:
\[
8x - 3x + 1 = 3x - 3x + 2
\]
\[
5x + 1 = 2
\]

2. Subtract 1 from both sides:
\[
5x + 1 - 1 = 2 - 1
\]
\[
5x = 1
\]

3. Divide by 5:
\[
x = \frac{1}{5}
\]

Solution: \( x = \frac{1}{5} \)

---

Equation 5: \( 7x - 10 = 3x - 8 \)

1. Subtract \( 3x \) from both sides:
\[
7x - 3x - 10 = 3x - 3x - 8
\]
\[
4x - 10 = -8
\]

2. Add 10 to both sides:
\[
4x - 10 + 10 = -8 + 10
\]
\[
4x = 2
\]

3. Divide by 4:
\[
x = \frac{2}{4} = \frac{1}{2}
\]

Solution: \( x = \frac{1}{2} \)

---

Equation 6: \( 6x - 12 = 2x - 6 \)

1. Subtract \( 2x \) from both sides:
\[
6x - 2x - 12 = 2x - 2x - 6
\]
\[
4x - 12 = -6
\]

2. Add 12 to both sides:
\[
4x - 12 + 12 = -6 + 12
\]
\[
4x = 6
\]

3. Divide by 4:
\[
x = \frac{6}{4} = \frac{3}{2}
\]

Solution: \( x = \frac{3}{2} \)

---

Equation 7: \( 4x - 23 = x - 7 \)

1. Subtract \( x \) from both sides:
\[
4x - x - 23 = x - x - 7
\]
\[
3x - 23 = -7
\]

2. Add 23 to both sides:
\[
3x - 23 + 23 = -7 + 23
\]
\[
3x = 16
\]

3. Divide by 3:
\[
x = \frac{16}{3}
\]

Solution: \( x = \frac{16}{3} \)

---

Equation 8: \( 8x - 8 = 3x - 2 \)

1. Subtract \( 3x \) from both sides:
\[
8x - 3x - 8 = 3x - 3x - 2
\]
\[
5x - 8 = -2
\]

2. Add 8 to both sides:
\[
5x - 8 + 8 = -2 + 8
\]
\[
5x = 6
\]

3. Divide by 5:
\[
x = \frac{6}{5}
\]

Solution: \( x = \frac{6}{5} \)

---

Equation 9: \( 11x + 7 = 6x + 7 \)

1. Subtract \( 6x \) from both sides:
\[
11x - 6x + 7 = 6x - 6x + 7
\]
\[
5x + 7 = 7
\]

2. Subtract 7 from both sides:
\[
5x + 7 - 7 = 7 - 7
\]
\[
5x = 0
\]

3. Divide by 5:
\[
x = 0
\]

Solution: \( x = 0 \)

---

Equation 10: \( 9x + 8 = 10 \)

1. Subtract 8 from both sides:
\[
9x + 8 - 8 = 10 - 8
\]
\[
9x = 2
\]

2. Divide by 9:
\[
x = \frac{2}{9}
\]

Solution: \( x = \frac{2}{9} \)

---

Equation 11: \( 5 + 3x = x + 8 \)

1. Subtract \( x \) from both sides:
\[
5 + 3x - x = x - x + 8
\]
\[
5 + 2x = 8
\]

2. Subtract 5 from both sides:
\[
5 + 2x - 5 = 8 - 5
\]
\[
2x = 3
\]

3. Divide by 2:
\[
x = \frac{3}{2}
\]

Solution: \( x = \frac{3}{2} \)

---

Equation 12: \( 4 + 7x = x + 5 \)

1. Subtract \( x \) from both sides:
\[
4 + 7x - x = x - x + 5
\]
\[
4 + 6x = 5
\]

2. Subtract 4 from both sides:
\[
4 + 6x - 4 = 5 - 4
\]
\[
6x = 1
\]

3. Divide by 6:
\[
x = \frac{1}{6}
\]

Solution: \( x = \frac{1}{6} \)

---

Equation 13: \( 6x - 8 = 4 - 3x \)

1. Add \( 3x \) to both sides:
\[
6x + 3x - 8 = 4 - 3x + 3x
\]
\[
9x - 8 = 4
\]

2. Add 8 to both sides:
\[
9x - 8 + 8 = 4 + 8
\]
\[
9x = 12
\]

3. Divide by 9:
\[
x = \frac{12}{9} = \frac{4}{3}
\]

Solution: \( x = \frac{4}{3} \)

---

Equation 14: \( 5x + 1 = 7 - 2x \)

1. Add \( 2x \) to both sides:
\[
5x + 2x + 1 = 7 - 2x + 2x
\]
\[
7x + 1 = 7
\]

2. Subtract 1 from both sides:
\[
7x + 1 - 1 = 7 - 1
\]
\[
7x = 6
\]

3. Divide by 7:
\[
x = \frac{6}{7}
\]

Solution: \( x = \frac{6}{7} \)

---

Equation 15: \( 16x - 3 = 1 - x \)

1. Add \( x \) to both sides:
\[
16x + x - 3 = 1 - x + x
\]
\[
17x - 3 = 1
\]

2. Add 3 to both sides:
\[
17x - 3 + 3 = 1 + 3
\]
\[
17x = 4
\]

3. Divide by 17:
\[
x = \frac{4}{17}
\]

Solution: \( x = \frac{4}{17} \)

---

Equation 16: \( -3x - 10 = 2x - 3 \)

1. Subtract \( 2x \) from both sides:
\[
-3x - 2x - 10 = 2x - 2x - 3
\]
\[
-5x - 10 = -3
\]

2. Add 10 to both sides:
\[
-5x - 10 + 10 = -3 + 10
\]
\[
-5x = 7
\]

3. Divide by -5:
\[
x = -\frac{7}{5}
\]

Solution: \( x = -\frac{7}{5} \)

---

Equation 17: \( 15x + 1 = 6 - 3x \)

1. Add \( 3x \) to both sides:
\[
15x + 3x + 1 = 6 - 3x + 3x
\]
\[
18x + 1 = 6
\]

2. Subtract 1 from both sides:
\[
18x + 1 - 1 = 6 - 1
\]
\[
18x = 5
\]

3. Divide by 18:
\[
x = \frac{5}{18}
\]

Solution: \( x = \frac{5}{18} \)

---

Equation 18: \( 11x - 20 = 10x - 15 \)

1. Subtract \( 10x \) from both sides:
\[
11x - 10x - 20 = 10x - 10x - 15
\]
\[
x - 20 = -15
\]

2. Add 20 to both sides:
\[
x - 20 + 20 = -15 + 20
\]
\[
x = 5
\]

Solution: \( x = 5 \)

---

Equation 19: \( 6 + 2x = 8 - 3x \)

1. Add \( 3x \) to both sides:
\[
6 + 2x + 3x = 8 - 3x + 3x
\]
\[
6 + 5x = 8
\]

2. Subtract 6 from both sides:
\[
6 + 5x - 6 = 8 - 6
\]
\[
5x = 2
\]

3. Divide by 5:
\[
x = \frac{2}{5}
\]

Solution: \( x = \frac{2}{5} \)

---

Equation 20: \( 7 + x = 9 - 5x \)

1. Add \( 5x \) to both sides:
\[
7 + x + 5x = 9 - 5x + 5x
\]
\[
7 + 6x = 9
\]

2. Subtract 7 from both sides:
\[
7 + 6x - 7 = 9 - 7
\]
\[
6x = 2
\]

3. Divide by 6:
\[
x = \frac{2}{6} = \frac{1}{3}
\]

Solution: \( x = \frac{1}{3} \)

---

Equation 21: \( 3y - 7 = y + 1 \)

1. Subtract \( y \) from both sides:
\[
3y - y - 7 = y - y + 1
\]
\[
2y - 7 = 1
\]

2. Add 7 to both sides:
\[
2y - 7 + 7 = 1 + 7
\]
\[
2y = 8
\]

3. Divide by 2:
\[
y = 4
\]

Solution: \( y = 4 \)

---

Final Answers:

\[
\boxed{
\begin{aligned}
1. & \ x = \frac{11}{4} \\
2. & \ x = \frac{5}{3} \\
3. & \ x = 2 \\
4. & \ x = \frac{1}{5} \\
5. & \ x = \frac{1}{2} \\
6. & \ x = \frac{3}{2} \\
7. & \ x = \frac{16}{3} \\
8. & \ x = \frac{6}{5} \\
9. & \ x = 0 \\
10. & \ x = \frac{2}{9} \\
11. & \ x = \frac{3}{2} \\
12. & \ x = \frac{1}{6} \\
13. & \ x = \frac{4}{3} \\
14. & \ x = \frac{6}{7} \\
15. & \ x = \frac{4}{17} \\
16. & \ x = -\frac{7}{5} \\
17. & \ x = \frac{5}{18} \\
18. & \ x = 5 \\
19. & \ x = \frac{2}{5} \\
20. & \ x = \frac{1}{3} \\
21. & \ y = 4 \\
\end{aligned}
}
\]