If |4x - 4| = 8 and |5y + 10| = 15, what is the smallest possible value of xy ?
﻿

-20

-15

-5

-1

We are given two absolute value equations to solve:

1. \(|4x - 4| = 8\)
2. \(|5y + 10| = 15\)

### Step 1: Solve \(|4x - 4| = 8\)
This absolute value equation splits into two possible cases:

- \(4x - 4 = 8\)
- \(4x - 4 = -8\)

#### Case 1: \(4x - 4 = 8\)
\[
4x = 12 \quad \Rightarrow \quad x = 3
\]

#### Case 2: \(4x - 4 = -8\)
\[
4x = -4 \quad \Rightarrow \quad x = -1
\]

So, the possible values of \(x\) are \(3\) and \(-1\).

### Step 2: Solve \(|5y + 10| = 15\)
This also splits into two cases:

- \(5y + 10 = 15\)
- \(5y + 10 = -15\)

#### Case 1: \(5y + 10 = 15\)
\[
5y = 5 \quad \Rightarrow \quad y = 1
\]

#### Case 2: \(5y + 10 = -15\)
\[
5y = -25 \quad \Rightarrow \quad y = -5
\]

So, the possible values of \(y\) are \(1\) and \(-5\).

### Step 3: Calculate the possible values of \(xy\)

We now combine the possible values of \(x\) and \(y\):

- \(x = 3\), \(y = 1 \quad \Rightarrow \quad xy = 3 \times 1 = 3\)
- \(x = 3\), \(y = -5 \quad \Rightarrow \quad xy = 3 \times (-5) = -15\)
- \(x = -1\), \(y = 1 \quad \Rightarrow \quad xy = (-1) \times 1 = -1\)
- \(x = -1\), \(y = -5 \quad \Rightarrow \quad xy = (-1) \times (-5) = 5\)

### Step 4: Find the smallest value of \(xy\)
From the calculated values, the smallest possible value of \(xy\) is \(-15\).

Thus, the correct answer is \(\boxed{-15}\).

Would you like further details or have any questions?

Here are 5 related questions to explore further:
1. How do you solve absolute value equations in general?
2. What happens if both sides of an absolute value equation contain variables?
3. How do you graph absolute value functions?
4. Can an absolute value equation ever have no solutions?
5. How would this problem change if \(xy\) had a specific constraint?

**Tip:** When solving absolute value equations, always consider both the positive and negative cases for the expression inside the absolute value.

If |4x - 4| = 8 and |5y + 10| = 15, what is the smallest possible value of xy?

Learn how to solve the absolute value equations |4x - 4| = 8 and |5y + 10| = 15 to find the smallest possible value of xy. This problem covers key algebraic concepts, including absolute value equations and their properties. Suitable for students in Grades 8-10.

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