Question

exercises 10-7

solve each inequality.

1. (4x > -32)
2. (8n - 13 \ge 11)
3. (\frac{w}{9} < 3)
4. (\frac{n}{2} + 3 < -1)
5. (x - 4 \le 16 + 5x)
6. (3(w - 1) > -12)
7. (9 > n + 10)
8. (4x + 8 \le 0)
9. (-7n \ge -21)
10. (18 - 3w < 6 - w)
11. (30 > \frac{n}{-2})
12. (9(n + 1) < -27)
13. (-x + 6 < 30)
14. (90 - n \ge -10n)
15. (-7x + 1 \ge -55)
16. (1 < \frac{2x}{8})

Explanation:

Solve inequalities 1 to 4

Using the Solving Linear Inequalities knowledge point:

1. For \(4x > -32\), divide both sides by \(4\):

$$x > -8$$

1. For \(8n - 13 \ge 11\), add \(13\) to both sides:

$$8n \ge 24$$

Divide by \(8\):

$$n \ge 3$$

1. For \(\frac{w}{9} < 3\), multiply both sides by \(9\):

$$w < 27$$

1. For \(\frac{n}{2} + 3 < -1\), subtract \(3\) from both sides:

$$\frac{n}{2} < -4$$

Multiply by \(2\):

$$n < -8$$

Solve inequalities 5 to 8

Using the Solving Linear Inequalities knowledge point:

1. For \(x - 4 \le 16 + 5x\), subtract \(x\) from both sides:

$$-4 \le 16 + 4x$$

Subtract \(16\) from both sides:

$$-20 \le 4x$$

Divide by \(4\):

$$-5 \le x \quad \text{or} \quad x \ge -5$$

1. For \(3(w - 1) > -12\), divide both sides by \(3\):

$$w - 1 > -4$$

Add \(1\) to both sides:

$$w > -3$$

1. For \(9 > n + 10\), subtract \(10\) from both sides:

$$-1 > n \quad \text{or} \quad n < -1$$

1. For \(4x + 8 \le 0\), subtract \(8\) from both sides:

$$4x \le -8$$

Divide by \(4\):

$$x \le -2$$

Solve inequalities 9 to 12

Using the Inequality Sign Reversal and Solving Linear Inequalities knowledge points:

1. For \(-7n \ge -21\), divide both sides by \(-7\) and reverse the inequality sign:

$$n \le 3$$

1. For \(18 - 3w < 6 - w\), add \(3w\) to both sides:

$$18 < 6 + 2w$$

Subtract \(6\) from both sides:

$$12 < 2w$$

Divide by \(2\):

$$6 < w \quad \text{or} \quad w > 6$$

1. For \(30 > \frac{n}{-2}\), multiply both sides by \(-2\) and reverse the inequality sign:

$$-60 < n \quad \text{or} \quad n > -60$$

1. For \(9(n + 1) < -27\), divide both sides by \(9\):

$$n + 1 < -3$$

Subtract \(1\) from both sides:

$$n < -4$$

Solve inequalities 13 to 16

Using the Inequality Sign Reversal and Solving Linear Inequalities knowledge points:

1. For \(-x + 6 < 30\), subtract \(6\) from both sides:

$$-x < 24$$

Multiply by \(-1\) and reverse the inequality sign:

$$x > -24$$

1. For \(90 - n \ge -10n\), add \(10n\) to both sides:

$$90 + 9n \ge 0$$

Subtract \(90\) from both sides:

$$9n \ge -90$$

Divide by \(9\):

$$n \ge -10$$

1. For \(-7x + 1 \ge -55\), subtract \(1\) from both sides:

$$-7x \ge -56$$

Divide by \(-7\) and reverse the inequality sign:

$$x \le 8$$

1. For \(1 < \frac{2x}{8}\), simplify the fraction:

$$1 < \frac{x}{4}$$

Multiply both sides by \(4\):

$$4 < x \quad \text{or} \quad x > 4$$

Answer:

No.	Answer
2	\(n \ge 3\)
3	\(w < 27\)
4	\(n < -8\)
5	\(x \ge -5\)
6	\(w > -3\)
7	\(n < -1\)
8	\(x \le -2\)
9	\(n \le 3\)
10	\(w > 6\)
11	\(n > -60\)
12	\(n < -4\)
13	\(x > -24\)
14	\(n \ge -10\)
15	\(x \le 8\)
16	\(x > 4\)