Determine the solution to the inequality.

|2x − 3| ≥ 9

 x ≤ −6 or x ≥ 6
 x ≤ −4 or x ≥ 6
 x ≤ −3 or x ≥ 3
 x ≤ −3 or x ≥ 6
Question 6(Multiple Choice Worth 1 points)
(05.02 MC)

Solve the system of equations using substitution.

y = 4x
y = 8x + 8

 (−3, −12)
 (−2, −8)
 (2, 8)
 (3, 12)
Question 7(Multiple Choice Worth 1 points)
(06.05 MC)

If f(x) = 3x + 1 and g(x) = x − 3, find the quantity f divided by g of 8.

 125
 25
 20
 5

Let's solve each problem one by one.

### Problem 1: Inequality \(|2x − 3| ≥ 9\)
To solve the absolute value inequality \(|2x − 3| ≥ 9\), we need to consider the two cases:

1. **Case 1: \(2x - 3 \geq 9\)**
   \[
   2x - 3 \geq 9
   \]
   \[
   2x \geq 12
   \]
   \[
   x \geq 6
   \]

2. **Case 2: \(2x - 3 \leq -9\)**
   \[
   2x - 3 \leq -9
   \]
   \[
   2x \leq -6
   \]
   \[
   x \leq -3
   \]

Thus, the solution to the inequality is:
\[
x \leq -3 \quad \text{or} \quad x \geq 6
\]
This corresponds to the answer **x ≤ −3 or x ≥ 6**.

### Problem 2: System of Equations
The system of equations is:
\[
y = 4x
\]
\[
y = 8x + 8
\]
We can solve this by substitution. Since both expressions equal \(y\), set them equal to each other:

\[
4x = 8x + 8
\]
Subtract \(8x\) from both sides:
\[
4x - 8x = 8
\]
\[
-4x = 8
\]
Divide by \(-4\):
\[
x = -2
\]
Now substitute \(x = -2\) back into the first equation to find \(y\):
\[
y = 4(-2) = -8
\]
Thus, the solution to the system is \((-2, -8)\), which corresponds to the answer **(−2, −8)**.

### Problem 3: Function Division
The functions are:
\[
f(x) = 3x + 1
\]
\[
g(x) = x - 3
\]
We need to find \(\frac{f(8)}{g(8)}\).

First, calculate \(f(8)\):
\[
f(8) = 3(8) + 1 = 24 + 1 = 25
\]

Next, calculate \(g(8)\):
\[
g(8) = 8 - 3 = 5
\]

Now, divide \(f(8)\) by \(g(8)\):
\[
\frac{f(8)}{g(8)} = \frac{25}{5} = 5
\]
The correct answer is **5**.

### Summary of Answers
1. **x ≤ −3 or x ≥ 6**
2. **(−2, −8)**
3. **5**

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Do you want more details or have any questions?

Here are some related questions to expand on:

1. How would you solve an absolute value inequality where the inequality is \(|2x − 5| < 7\)?
2. Can you solve the system of equations \(y = 2x + 1\) and \(y = -3x + 4\)?
3. How do you solve inequalities involving absolute values with quadratic expressions?
4. What is the solution to the system of equations using the elimination method for \(y = 3x\) and \(2y = 6x + 4\)?
5. How would you find the intersection point of the equations \(y = x + 2\) and \(y = -x + 4\)?
6. Can you find the value of \(x\) where \(f(x)\) and \(g(x)\) intersect if \(f(x) = 5x + 2\) and \(g(x) = -2x + 7\)?
7. How would you solve for \(x\) in the inequality \(3|x - 2| + 4 > 10\)?
8. What is the value of \(f \div g\) when \(x = 10\) for \(f(x) = 4x - 5\) and \(g(x) = x + 7\)?

**Tip:** For absolute value inequalities, always split the inequality into two cases—one positive and one negative.

Learn how to solve an absolute value inequality |2x - 3| ≥ 9, a system of equations using substitution, and function division f(x) ÷ g(x) = 8 for f(x) = 3x + 1 and g(x) = x - 3. Detailed step-by-step solutions and explanations provided.

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