Question

1. $10 geq \frac{x}{2} + 8$
2. $21 < -9 + 10x$
3. $3(-3 + 7x) \leq -93$
4. $-6(1 + 8x) \geq 138$
5. $86 \leq -4(1 - 5x) - 2x$
6. $102 > 6x + 3(4 - 8x)$
7. $-6(-3 + 6x) > -38 - 8x$
8. $7(6x - 4) \geq 6x - 28$
9. $8(4x + 3) \geq -1 + 7x$
10. $8x + 40 > -8(1 - 7x)$

word bank
(number lines with various labels and arrows)

Explanation:

Step1: Solve the inequality \(10 \geq \frac{x}{2} + 8\)

Subtract 8 from both sides: \(10 - 8 \geq \frac{x}{2} + 8 - 8\)
Simplify: \(2 \geq \frac{x}{2}\)
Multiply both sides by 2: \(2\times2 \geq \frac{x}{2}\times2\)
Result: \(4 \geq x\) or \(x \leq 4\)

Step2: Solve the inequality \(21 < -9 + 10x\)

Add 9 to both sides: \(21 + 9 < -9 + 10x + 9\)
Simplify: \(30 < 10x\)
Divide both sides by 10: \(\frac{30}{10} < \frac{10x}{10}\)
Result: \(3 < x\) or \(x > 3\)

Step3: Solve the inequality \(3(-3 + 7x) \leq -93\)

Distribute the 3: \( -9 + 21x \leq -93\)
Add 9 to both sides: \( -9 + 21x + 9 \leq -93 + 9\)
Simplify: \(21x \leq -84\)
Divide both sides by 21: \(\frac{21x}{21} \leq \frac{-84}{21}\)
Result: \(x \leq -4\)

Step4: Solve the inequality \(-6(1 + 8x) \geq 138\)

Distribute the -6: \( -6 - 48x \geq 138\)
Add 6 to both sides: \( -6 - 48x + 6 \geq 138 + 6\)
Simplify: \( -48x \geq 144\)
Divide both sides by -48 (reverse inequality): \(\frac{-48x}{-48} \leq \frac{144}{-48}\)
Result: \(x \leq -3\)

Step5: Solve the inequality \(86 \leq -4(1 - 5x) - 2x\)

Distribute the -4: \(86 \leq -4 + 20x - 2x\)
Combine like terms: \(86 \leq -4 + 18x\)
Add 4 to both sides: \(86 + 4 \leq -4 + 18x + 4\)
Simplify: \(90 \leq 18x\)
Divide both sides by 18: \(\frac{90}{18} \leq \frac{18x}{18}\)
Result: \(5 \leq x\) or \(x \geq 5\)

Step6: Solve the inequality \(102 > 6x + 3(4 - 8x)\)

Distribute the 3: \(102 > 6x + 12 - 24x\)
Combine like terms: \(102 > -18x + 12\)
Subtract 12 from both sides: \(102 - 12 > -18x + 12 - 12\)
Simplify: \(90 > -18x\)
Divide both sides by -18 (reverse inequality): \(\frac{90}{-18} < \frac{-18x}{-18}\)
Result: \(-5 < x\) or \(x > -5\)

Step7: Solve the inequality \(-6(-3 + 6x) > -38 - 8x\)

Distribute the -6: \(18 - 36x > -38 - 8x\)
Add 36x to both sides: \(18 - 36x + 36x > -38 - 8x + 36x\)
Simplify: \(18 > -38 + 28x\)
Add 38 to both sides: \(18 + 38 > -38 + 28x + 38\)
Simplify: \(56 > 28x\)
Divide both sides by 28: \(\frac{56}{28} > \frac{28x}{28}\)
Result: \(2 > x\) or \(x < 2\)

Step8: Solve the inequality \(7(6x - 4) \geq 6x - 28\)

Distribute the 7: \(42x - 28 \geq 6x - 28\)
Subtract 6x from both sides: \(42x - 28 - 6x \geq 6x - 28 - 6x\)
Simplify: \(36x - 28 \geq -28\)
Add 28 to both sides: \(36x - 28 + 28 \geq -28 + 28\)
Simplify: \(36x \geq 0\)
Divide both sides by 36: \(\frac{36x}{36} \geq \frac{0}{36}\)
Result: \(x \geq 0\)

Step9: Solve the inequality \(8(4x + 3) \geq -1 + 7x\)

Distribute the 8: \(32x + 24 \geq -1 + 7x\)
Subtract 7x from both sides: \(32x + 24 - 7x \geq -1 + 7x - 7x\)
Simplify: \(25x + 24 \geq -1\)
Subtract 24 from both sides: \(25x + 24 - 24 \geq -1 - 24\)
Simplify: \(25x \geq -25\)
Divide both sides by 25: \(\frac{25x}{25} \geq \frac{-25}{25}\)
Result: \(x \geq -1\)

Step10: Solve the inequality \(8x + 40 > -8(1 - 7x)\)

Distribute the -8: \(8x + 40 > -8 + 56x\)
Subtract 8x from both sides: \(8x + 40 - 8x > -8 + 56x - 8x\)
Simplify: \(40 > -8 + 48x\)
Add 8 to both sides: \(40 + 8 > -8 + 48x + 8\)
Simplify: \(48 > 48x\)
Divide both sides by 48: \(\frac{48}{48} > \frac{48x}{48}\)
Result: \(1 > x\) or \(x < 1\)

Answer:

1. \(x \leq 4\)
2. \(x > 3\)
3. \(x \leq -4\)
4. \(x \leq -3\)
5. \(x \geq 5\)
6. \(x > -5\)
7. \(x < 2\)
8. \(x \geq 0\)
9. \(x \geq -1\)
10. \(x < 1\)