Question

solving and graphing linear inequalities worksheet
solve each inequality. sketch the solution set

1. $\frac{x}{4}geq2 - 4$
2. $3a + 7leq16$
3. $16<3a + 4$
4. $20-8n>7n$
5. $3xgeq - 9$
6. $4b+8leq24$
7. $0.7m + 0.3mgeq2m - 4$
8. $4(5x + 7)leq13$
9. $1.7y-0.78>5$
10. $7(7a - 9)leq84$
11. $3(9z + 4)>35z - 4$
12. $5(12 - 3n)<165$
13. $4x - 2>-7(4x - 2)$
14. $\frac{1}{3}(2y - 3)>y + 2$
15. $25d+15leq75$

Explanation:

Problem 1: Solve \( \frac{x}{2} \geq -4 \)

Step1: Multiply both sides by 2

\( x \geq -8 \)

Problem 2: Solve \( 3a + 7 \leq 16 \)

Step1: Subtract 7 from both sides

\( 3a \leq 9 \)

Step2: Divide by 3

\( a \leq 3 \)

Problem 3: Solve \( 16 < 3q + 4 \)

Step1: Subtract 4 from both sides

\( 12 < 3q \)

Step2: Divide by 3

\( 4 < q \) (or \( q > 4 \))

Problem 4: Solve \( 20 - 8n > 7n \)

Step1: Add 8n to both sides

\( 20 > 15n \)

Step2: Divide by 15

\( \frac{4}{3} > n \) (or \( n < \frac{4}{3} \))

Problem 5: Solve \( 3x \geq -9 \)

Step1: Divide by 3

\( x \geq -3 \)

Problem 6: Solve \( 4b + 9 \leq 7 \)

Step1: Subtract 9 from both sides

\( 4b \leq -2 \)

Step2: Divide by 4

\( b \leq -\frac{1}{2} \)

Problem 7: Solve \( 0.7m + 0.3m \geq 2m - 4 \)

Step1: Combine like terms

\( m \geq 2m - 4 \)

Step2: Subtract 2m from both sides

\( -m \geq -4 \)

Step3: Multiply by -1 (reverse inequality)

\( m \leq 4 \)

Problem 8: Solve \( 4(5x + 7) \leq 13 \)

Step1: Expand left side

\( 20x + 28 \leq 13 \)

Step2: Subtract 28 from both sides

\( 20x \leq -15 \)

Step3: Divide by 20

\( x \leq -\frac{3}{4} \)

Problem 9: Solve \( 1.7y - 0.78 > 5 \)

Step1: Add 0.78 to both sides

\( 1.7y > 5.78 \)

Step2: Divide by 1.7

\( y > 3.4 \)

Problem 10: Solve \( 7(7a - 9) \leq 84 \)

Step1: Divide by 7

\( 7a - 9 \leq 12 \)

Step2: Add 9 to both sides

\( 7a \leq 21 \)

Step3: Divide by 7

\( a \leq 3 \)

Problem 11: Solve \( 3(9z + 4) > 35z - 4 \)

Step1: Expand left side

\( 27z + 12 > 35z - 4 \)

Step2: Subtract 27z from both sides

\( 12 > 8z - 4 \)

Step3: Add 4 to both sides

\( 16 > 8z \)

Step4: Divide by 8

\( 2 > z \) (or \( z < 2 \))

Problem 12: Solve \( 5(12 - 3n) < 165 \)

Step1: Divide by 5

\( 12 - 3n < 33 \)

Step2: Subtract 12 from both sides

\( -3n < 21 \)

Step3: Divide by -3 (reverse inequality)

\( n > -7 \)

Problem 13: Solve \( 4x - 2 > -7(4x - 2) \)

Step1: Expand right side

\( 4x - 2 > -28x + 14 \)

Step2: Add 28x to both sides

\( 32x - 2 > 14 \)

Step3: Add 2 to both sides

\( 32x > 16 \)

Step4: Divide by 32

\( x > \frac{1}{2} \)

Problem 14: Solve \( \frac{1}{3}(2y - 3) > y + 2 \)

Step1: Multiply by 3

\( 2y - 3 > 3y + 6 \)

Step2: Subtract 2y from both sides

\( -3 > y + 6 \)

Step3: Subtract 6 from both sides

\( -9 > y \) (or \( y < -9 \))

Problem 15: Solve \( 2.5d + 15 \leq 75 \)

Step1: Subtract 15 from both sides

\( 2.5d \leq 60 \)

Step2: Divide by 2.5

\( d \leq 24 \)

Answer:

1. \( x \geq -8 \) (Sketch: Closed circle at -8, arrow to the right)
2. \( a \leq 3 \) (Sketch: Closed circle at 3, arrow to the left)
3. \( q > 4 \) (Sketch: Open circle at 4, arrow to the right)
4. \( n < \frac{4}{3} \) (Sketch: Open circle at \( \frac{4}{3} \), arrow to the left)
5. \( x \geq -3 \) (Sketch: Closed circle at -3, arrow to the right)
6. \( b \leq -\frac{1}{2} \) (Sketch: Closed circle at \( -\frac{1}{2} \), arrow to the left)
7. \( m \leq 4 \) (Sketch: Closed circle at 4, arrow to the left)
8. \( x \leq -\frac{3}{4} \) (Sketch: Closed circle at \( -\frac{3}{4} \), arrow to the left)
9. \( y > 3.4 \) (Sketch: Open circle at 3.4, arrow to the right)
10. \( a \leq 3 \) (Sketch: Closed circle at 3, arrow to the left)
11. \( z < 2 \) (Sketch: Open circle at 2, arrow to the left)
12. \( n > -7 \) (Sketch: Open circle at -7, arrow to the right)
13. \( x > \frac{1}{2} \) (Sketch: Open circle at \( \frac{1}{2} \), arrow to the right)
14. \( y < -9 \) (Sketch: Open circle at -9, arrow to the left)
15. \( d \leq 24 \) (Sketch: Closed circle at 24, arrow to the left)