Question

directions: solve each inequality and graph the solution on the number line.

1. $6(x - 1) \leq -30$
2. $-2(p - 7) < -8$
3. $-2(4a + 4) - 5a > -35$
4. $29 - 2(3 - 5w) \geq 13$
5. $3k + 23 < 7k - 41$
6. $10 - r \leq -25 - 6r$
7. $-2(5v - 12) > 6 - v$
8. $4(3m + 10) \geq -3m - 51 + 8m$

Explanation:

Step1: Solve \(6(x - 1)\leq - 30\)

Divide both sides by 6: \(x - 1\leq\frac{-30}{6}=-5\)
Add 1 to both sides: \(x\leq - 5 + 1=-4\)

Step2: Solve \(-2(p - 7)\lt - 8\)

Divide both sides by - 2 (reverse inequality): \(p - 7>\frac{-8}{-2}=4\)
Add 7 to both sides: \(p>4 + 7 = 11\)

Step3: Solve \(-2(4a + 4)-5a>-35\)

Expand: \(-8a-8 - 5a>-35\)
Combine like terms: \(-13a-8>-35\)
Add 8: \(-13a>-35 + 8=-27\)
Divide by - 13 (reverse inequality): \(a<\frac{-27}{-13}=\frac{27}{13}\approx2.08\)

Step4: Solve \(29-2(3 - 5w)\geq13\)

Expand: \(29-6 + 10w\geq13\)
Simplify: \(23 + 10w\geq13\)
Subtract 23: \(10w\geq13 - 23=-10\)
Divide by 10: \(w\geq - 1\)

Step5: Solve \(3k + 23\lt7k-41\)

Subtract \(3k\): \(23\lt4k-41\)
Add 41: \(64\lt4k\)
Divide by 4: \(16\lt k\) or \(k > 16\)

Step6: Solve \(10-r\leq - 25-6r\)

Add \(6r\): \(10 + 5r\leq - 25\)
Subtract 10: \(5r\leq - 25-10=-35\)
Divide by 5: \(r\leq - 7\)

Step7: Solve \(-2(5v - 12)>6 - v\)

Expand: \(-10v + 24>6 - v\)
Add \(10v\): \(24>6 + 9v\)
Subtract 6: \(18>9v\)
Divide by 9: \(2>v\) or \(v < 2\)

Step8: Solve \(4(3m + 10)\geq - 3m-51 + 8m\)

Expand: \(12m+40\geq5m - 51\)
Subtract \(5m\): \(7m+40\geq - 51\)
Subtract 40: \(7m\geq - 51-40=-91\)
Divide by 7: \(m\geq\frac{-91}{7}=-13\)

Answer:

1. \(x\leq - 4\)
2. \(p > 11\)
3. \(a<\frac{27}{13}\)
4. \(w\geq - 1\)
5. \(k > 16\)
6. \(r\leq - 7\)
7. \(v < 2\)
8. \(m\geq - 13\)