Question

solving inequalities

solve and graph the given inequalities on a number line.

1. \\(4(2x - 3) \ge -8\\)
2. \\(4x + 6 \le 18\\)
3. \\(-10 < 6x + 2\\)
4. \\(4(x - 2) > 20\\)
5. \\(6x + 7 < 37\\)
6. \\(6 - 5x > -10\\)
7. \\(7 + 4x \ge 19\\)
8. \\(5y < 8 - 7\\)
9. \\(6x - 7 \le 5\\)
10. \\(6x + 2 \le 32\\)

Explanation:

Identify the given inequalities

We are given 10 linear inequalities to solve.

1. \(4(2x - 3) \ge -8\)
2. \(4x + 6 \le 18\)
3. \(-10 < 6x + 2\)
4. \(4(x - 2) > 20\)
5. \(6x + 7 < 37\)
6. \(6 - 5x > -10\)
7. \(7 + 4x \ge 19\)
8. \(5y < 8 - 7\)
9. \(6x - 7 \le 5\)
10. \(6x + 2 \le 32\)

Solve inequalities 1 to 5

Using the Basic Addition and Basic Subtraction knowledge points:

• Inequality 1:

Divide by 4: \(2x - 3 \ge -2\)
Add 3: \(2x \ge 1\)
Divide by 2: \(x \ge \frac{1}{2}\)

• Inequality 2:

Subtract 6: \(4x \le 12\)
Divide by 4: \(x \le 3\)

• Inequality 3:

Subtract 2: \(-12 < 6x\)
Divide by 6: \(-2 < x\) or \(x > -2\)

• Inequality 4:

Divide by 4: \(x - 2 > 5\)
Add 2: \(x > 7\)

• Inequality 5:

Subtract 7: \(6x < 30\)
Divide by 6: \(x < 5\)

Solve inequalities 6 to 10

Using the Basic Addition and Basic Subtraction knowledge points:

• Inequality 6:

Subtract 6: \(-5x > -16\)
Divide by -5 (reverse inequality sign): \(x < \frac{16}{5}\) (or \(x < 3.2\))

• Inequality 7:

Subtract 7: \(4x \ge 12\)
Divide by 4: \(x \ge 3\)

• Inequality 8:

Simplify right side: \(5y < 1\)
Divide by 5: \(y < \frac{1}{5}\) (or \(y < 0.2\))

• Inequality 9:

Add 7: \(6x \le 12\)
Divide by 6: \(x \le 2\)

• Inequality 10:

Subtract 2: \(6x \le 30\)
Divide by 6: \(x \le 5\)

Answer:

No.	Answer
2	\(x \le 3\)
3	\(x > -2\)
4	\(x > 7\)
5	\(x < 5\)
6	\(x < \frac{16}{5}\)
7	\(x \ge 3\)
8	\(y < \frac{1}{5}\)
9	\(x \le 2\)
10	\(x \le 5\)