Edia | Free math homework in minutes - Free Printable

Let’s solve each inequality one by one, step by step. We’ll simplify both sides, move variables to one side and constants to the other, then solve for the variable. Since graphing is requested but we can’t draw here, I’ll describe how to graph it (open/closed circle and direction).

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Problem 1: Solve for m

Original:
36 – 34 + 9m – 32 > 11m – 12m

Step 1: Simplify both sides.

Left: 36 – 34 = 2; 2 – 32 = -30 → so left is: -30 + 9m
Right: 11m – 12m = -m

So:
-30 + 9m > -m

Step 2: Add m to both sides to get all m terms on left.

-30 + 9m + m > 0
→ -30 + 10m > 0

Step 3: Add 30 to both sides.

10m > 30

Step 4: Divide by 10.

m > 3

Graph: Open circle at 3, arrow pointing right.

✔ Final Answer for #1: m > 3

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Problem 2: Solve for a

Original:
5a – 3a – 7 ≥ 17 – 12 + 6a

Step 1: Simplify both sides.

Left: 5a – 3a = 2a → 2a – 7
Right: 17 – 12 = 5 → 5 + 6a

So:
2a – 7 ≥ 5 + 6a

Step 2: Subtract 6a from both sides.

2a – 6a – 7 ≥ 5
→ -4a – 7 ≥ 5

Step 3: Add 7 to both sides.

-4a ≥ 12

Step 4: Divide by -4 → REMEMBER: flipping inequality when dividing by negative!

a ≤ -3

Graph: Closed circle at -3, arrow pointing left.

✔ Final Answer for #2: a ≤ -3

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Problem 3: Solve for x

Original:
10 + 6x – 14 – 3x > -13 – 6x

Step 1: Simplify left side.

10 – 14 = -4; 6x – 3x = 3x → so: -4 + 3x
Right: -13 – 6x

So:
-4 + 3x > -13 – 6x

Step 2: Add 6x to both sides.

-4 + 9x > -13

Step 3: Add 4 to both sides.

9x > -9

Step 4: Divide by 9.

x > -1

Graph: Open circle at -1, arrow right.

✔ Final Answer for #3: x > -1

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Problem 4: Solve for x

Original:
2x – 6x + 5 ≤ 1 – 3x + 3

Step 1: Simplify both sides.

Left: 2x – 6x = -4x → -4x + 5
Right: 1 + 3 = 4 → 4 – 3x

So:
-4x + 5 ≤ 4 – 3x

Step 2: Add 4x to both sides.

5 ≤ 4 + x

Step 3: Subtract 4 from both sides.

1 ≤ x → or x ≥ 1

Graph: Closed circle at 1, arrow right.

✔ Final Answer for #4: x ≥ 1

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Problem 5: Solve for a

Original:
10a > 4(-6a – 21) + 26a + 60

Step 1: Distribute the 4 on right.

4 × (-6a) = -24a; 4 × (-21) = -84 → so: -24a – 84 + 26a + 60

Combine like terms on right:
-24a + 26a = 2a; -84 + 60 = -24 → so right is: 2a – 24

Now:
10a > 2a – 24

Step 2: Subtract 2a from both sides.

8a > -24

Step 3: Divide by 8.

a > -3

Graph: Open circle at -3, arrow right.

✔ Final Answer for #5: a > -3

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Problem 6: Solve for n

Original:
3n > 4(-6n – 11) + 32 + 23n

Step 1: Distribute 4.

4 × (-6n) = -24n; 4 × (-11) = -44 → so: -24n – 44 + 32 + 23n

Combine like terms:
-24n + 23n = -n; -44 + 32 = -12 → right is: -n – 12

Now:
3n > -n – 12

Step 2: Add n to both sides.

4n > -12

Step 3: Divide by 4.

n > -3

Graph: Open circle at -3, arrow right.

✔ Final Answer for #6: n > -3

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Problem 7: Solve for y

Original:
-27 – 6y > 2(6y – 27) – 9y

Step 1: Distribute 2 on right.

2 × 6y = 12y; 2 × (-27) = -54 → so: 12y – 54 – 9y

Combine like terms: 12y – 9y = 3y → right is: 3y – 54

Now:
-27 – 6y > 3y – 54

Step 2: Add 6y to both sides.

-27 > 9y – 54

Step 3: Add 54 to both sides.

27 > 9y

Step 4: Divide by 9.

3 > y → or y < 3

Graph: Open circle at 3, arrow left.

✔ Final Answer for #7: y < 3

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Problem 8: Solve for y

Original:
-(12/5)y – 59/20 – 9/4 > -2y

First, let’s write all terms clearly.

Left: -(12/5)y – 59/20 – 9/4
Right: -2y

Step 1: Combine constants on left. Need common denominator for 20 and 4 → LCD = 20.

9/4 = 45/20

So: -59/20 – 45/20 = -104/20 = -26/5

So left becomes: -(12/5)y – 26/5

Equation now:
-(12/5)y – 26/5 > -2y

Step 2: Let’s eliminate fractions by multiplying entire inequality by 5.

5 × [-(12/5)y] = -12y
5 × [-26/5] = -26
5 × [-2y] = -10y

So:
-12y – 26 > -10y

Step 3: Add 12y to both sides.

-26 > 2y

Step 4: Divide by 2.

-13 > y → or y < -13

Graph: Open circle at -13, arrow left.

✔ Final Answer for #8: y < -13

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Problem 9: Solve for z

Original:
(16/9)z + 47/12 – 5/4 ≥ (14/9)z

Step 1: Move variable terms to one side. Subtract (14/9)z from both sides.

[(16/9)z – (14/9)z] + 47/12 – 5/4 ≥ 0
→ (2/9)z + 47/12 – 5/4 ≥ 0

Step 2: Combine constants. LCD of 12 and 4 is 12.

5/4 = 15/12

So: 47/12 – 15/12 = 32/12 = 8/3

Now:
(2/9)z + 8/3 ≥ 0

Step 3: Subtract 8/3 from both sides.

(2/9)z ≥ -8/3

Step 4: Multiply both sides by reciprocal of 2/9 → which is 9/2.

z ≥ (-8/3) × (9/2)

Multiply: (-8 × 9) / (3 × 2) = -72 / 6 = -12

So: z ≥ -12

Graph: Closed circle at -12, arrow right.

✔ Final Answer for #9: z ≥ -12

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Problem 10: Solve for k

Original:
(1/2)k – 55/6 + 14/3 > k

Step 1: Combine constants on left. LCD of 6 and 3 is 6.

14/3 = 28/6

So: -55/6 + 28/6 = -27/6 = -9/2

Now:
(1/2)k – 9/2 > k

Step 2: Subtract (1/2)k from both sides.

-9/2 > k – (1/2)k
→ -9/2 > (1/2)k

Step 3: Multiply both sides by 2.

-9 > k → or k < -9

Graph: Open circle at -9, arrow left.

✔ Final Answer for #10: k < -9

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Final Answers:

1. m > 3
2. a ≤ -3
3. x > -1
4. x ≥ 1
5. a > -3
6. n > -3
7. y < 3
8. y < -13
9. z ≥ -12
10. k < -9