SOLVING TWO-STEP AND MULTI-STEP INEQUALITIES WORKSHEET

Problems 1-8 : Solve each inequality and graph the solutions : 

Problem 1 : 

170 + 4x ≤ 510

Problem 2 : 

y/2 + 2 > 4.25

Problem 3 : 

8 - 2z ≤ 22

Problem 4 : 

(7x + 2)/5 < 8.8

Problem 5 : 

(1 -3y)/2 ≤ -2.5

Problem 6 : 

-5 + (-7) < -5x - 2

Problem 7 : 

-3(3 - y) < 16

Problem 8 : 

4m/5 + 1/2 < 3/5

Problem 9 : 

To win the blue ribbon for the Heaviest Pumpkin Crop at the county fair, the average weight of David's two pumpkins must be greater than 829 lb. One of his pumpkins weighs 897 lb. What is the least number  of pounds the second pumpkin could weigh in order for David to win the blue ribbon?

Problem 10 : 

The average of Ryan’s two test scores must be at least 90 to make an A in the class. Ryan got a 95 on his first test. What scores can Ryan get on his second test to make an A in the class?

Detailed Answer Key

1. Answer :

170 + 4x ≤ 510

Because 170 is added to 4x, subtract 160 from each side to undo the addition.

(170 + 4x) - 170 ≤ 510 - 170

170 + 4x - 170 ≤ 340

4x ≤ 340

Because x is multiplied by 4, divide each side by 4 to undo the multiplication.

4x/4 ≤ 340/4

x ≤ 85

2. Answer :

y/2 + 2 > 4.25

Because 2 is added to y/2, subtract 2 from each side to undo the addition.

(y/2 + 2) - 2 > 4.25 - 2

y/2 + 2 - 2 > 2.25

y/2 > 2.25

Because y is divided by 2, multiply each side by 2 to undo the division.

2(y/2) > 2(2.25)

y > 4.5

3. Answer :

8 - 2z ≤ 22

Because 8 is added to -2z, subtract 8 from each side to undo the addition.

(8 - 2z) - 8 ≤ 22 - 8

8 - 2z - 8 ≤ 14

-2z ≤ 14

Because z is multiplied by -2, divide each side by -2 to undo the multiplication and change ≤ to ≥.

-2z/(-2) ≥ 14/(-2)

z ≥ -7

4. Answer :

(7x + 2)/5 < 8.8

Because (7x + 2) is divided by 5, multiply each side by 5 to undo the division.

5[(7x + 2)/5] < 5(8.8)

7x + 2 < 44

Because 2 is added to 7x, subtract 2 from each side to undo the addition.

(7x + 2) - 2 < 44 - 2

7x + 2 - 2 < 44 - 2

7x < 42

Because x is multiplied by 7, divide each side by 7 to undo the multiplication.

7x/7 < 42/7

x < 6

x < 6

5. Answer :

(1 -3y)/2 ≤ -2.5  

Because (1 - 3y) is divided by 2, multiply each side by 2 to undo the division.

2[(1 - 3y)/2] ≤ 2(-2.5)

1 - 3y ≤ -5

Because 1 is added to -3y, subtract 1 from each side to undo the addition.

(1 -3y) - 1 ≤ -5 - 1

1 - 3y - 1 ≤ -6

-3y ≤ -6

Because y is multiplied by -3, divide each side by -3 and change ≤ to ≥.

-3y/(-3) ≥ -6/(-3)

y ≥ 2

6. Answer :

-5 + (-7) < -5x - 2

Combine like terms. 

-12 < -5x - 2

Because 2 is subtracted from -5x, add 2 to each side to undo the subtraction.

-12 + 2 < (-5x - 2) + 2

-10 < -5x - 2 + 2

-10 < -5x

Because x is multiplied by -5, divide each side by -5 to undo the multiplication and change < to >.

-10/(-5) > -5x/(-5)

2 > x

x < 2

7. Answer :

-3(3 - y) < 16

Distribute -3 on the left side.

-9 + 3y < 16

Because -9 is added to 3y, add 9 to each side to undo the addition.

9 + (-9 + 3y) < 9 + 16

9 - 9 + 3y < 25

3y < 25

Because y is multiplied by 3, divide each side by 3 to undo the multiplication.

3y/3 < 25/3

y < 8⅓

8. Answer :

4m/5 + 1/2 < 3/5

In the fractions above, we find the denominators 2 and 5. 

Least common multiple of the denominators (2, 5) = 10.

Multiply each side by the least common multiple, 10 to get rid of the fractions. 

10(4m/5 + 1/2) < 10(3/5)

10(4m/5) + 10(1/2) < 30/5

40m/5 + 10/2 < 6

8m + 5 < 6

Because 5 is added to 8m, subtract 5 to each side to undo the addition.

(8m + 5) - 5 < 6 - 5

8m + 5 - 5 < 1

8m < 1

Because x is multiplied by 8, divide each side by 8 to undo the multiplication.

8m/8 < 1/8

m < 1/8

9. Answer  :

Let x represent the weight of the second pumpkin. The average weight of the pumpkins is the sum of each weight divided by 2.

(897 + x) divided by 2 must be greater than 829

(897 + x)/2 > 829

Because (897 + x) is divided by 2, multiply each side by 2 to undo the division.

2[(897 + x)/2] > 2(829)

897 + x > 1658

Because 897 is added to x, subtract 897 from each side to undo the addition.

(897 + x) - 897 > 1658 - 897

897 + x - 897 > 761

x > 761

The second pumpkin must weigh more than 761 pounds.

10. Answer  :

Let y represent the score in the second test. The average of score is the sum of the two scores divided by 2.

(95 + y) divided by 2 must be greater than or equal to 90

(95 + y)/2 ≥ 90

Because (95 + y) is divided by 2, multiply each side by 2 to undo the division.

2[(95 + y)/2] ≥ 2(90)

95 + y ≥ 180

Because 95 is added to y, subtract 95 from each side to undo the addition.

(95 + y) - 95 ≥ 180 - 95

95 + y - 95 ≥ 85

y ≥ 85

Ryan has to get at least 85 on his second test to make an A in the class.

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