https://file.aiquickdraw.com/mathbot/file/1726533510950rn1xec9q.jpg

Let's break down and solve the inequalities from the image one by one:

### Problem 5:
Solve the inequality \( 5x - 3(x - 6) \geq 0 \).

1. Distribute the \( -3 \) to the terms inside the parentheses:
   \[
   5x - 3(x - 6) = 5x - 3x + 18
   \]

2. Combine like terms:
   \[
   5x - 3x + 18 = 2x + 18
   \]

3. Now solve the inequality \( 2x + 18 \geq 0 \):
   \[
   2x \geq -18
   \]

4. Divide by 2:
   \[
   x \geq -9
   \]

### Problem 6:
Solve \( 3x + 2(4x + 2) \leq 2(6x + 1) \).

1. Distribute the terms on both sides:
   \[
   3x + 8x + 4 \leq 12x + 2
   \]

2. Simplify:
   \[
   11x + 4 \leq 12x + 2
   \]

3. Move the variables to one side:
   \[
   11x - 12x \leq 2 - 4
   \]

4. Simplify:
   \[
   -x \leq -2
   \]

5. Divide by -1 and reverse the inequality:
   \[
   x \geq 2
   \]

### Problem 7:
Solve \( 3(x + 1) + 11 < -2(x + 13) \).

1. Distribute on both sides:
   \[
   3x + 3 + 11 < -2x - 26
   \]

2. Simplify:
   \[
   3x + 14 < -2x - 26
   \]

3. Move the variables to one side:
   \[
   3x + 2x < -26 - 14
   \]

4. Simplify:
   \[
   5x < -40
   \]

5. Divide by 5:
   \[
   x < -8
   \]

### Problem 8:
Solve \( x \leq \frac{6x + 3}{5} \).

1. Multiply both sides by 5 to eliminate the denominator:
   \[
   5x \leq 6x + 3
   \]

2. Move the terms to one side:
   \[
   5x - 6x \leq 3
   \]

3. Simplify:
   \[
   -x \leq 3
   \]

4. Divide by -1 and reverse the inequality:
   \[
   x \geq -3
   \]

---

### Word Problems (9 & 10):
For problems 9 and 10, we can write the inequalities based on the descriptions.

### Problem 9:
"A number is less than one fourth the sum of three times the number and four."

Let the number be \( x \).

1. The inequality is:
   \[
   x < \frac{1}{4}(3x + 4)
   \]

### Problem 10:
"Two times the sum of a number and four is no more than three times the sum of the number and seven decreased by four."

1. The inequality is:
   \[
   2(x + 4) \leq 3(x + 7) - 4
   \]

Would you like to proceed with solving the inequalities for problems 9 and 10 or have further clarifications on any specific step?

Here are some follow-up questions to deepen your understanding:

1. How would you handle solving inequalities that involve fractions?
2. What is the impact of multiplying both sides of an inequality by a negative number?
3. Can you explain why we reverse the inequality sign when dividing by a negative?
4. How would you write an inequality from a verbal description in problem 9?
5. What strategies do you use to simplify complex expressions involving variables?

**Tip**: Always remember to reverse the inequality sign when multiplying or dividing by a negative number!

Solve the inequalities and write inequalities for word problems involving a number.

Learn how to solve algebraic inequalities involving distribution and combining like terms. This lesson also includes solving word problems by translating them into inequalities. Suitable for students in Grades 8-10.

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