The given equation is:

$|3x + 5| + |2x - 3| = 25$

We need to solve this absolute value equation and then find the sum of the highest and lowest $x$-values.

Step-by-Step Solution

To solve this, we break it into cases based on when the expressions inside the absolute value change their signs.

Case 1: $3x + 5 \geq 0$ and $2x - 3 \geq 0$

In this case, both absolute values are positive. Therefore, the equation becomes:

$(3x + 5) + (2x - 3) = 25$

Simplifying this:

$5x + 2 = 25$

$5x = 23 \quad \Rightarrow \quad x = \frac{23}{5} = 4.6$

Case 2: $3x + 5 \geq 0$ and $2x - 3 < 0$

Here, $3x + 5$ is positive, but $2x - 3$ is negative. Thus, we rewrite the equation as:

$(3x + 5) - (2x - 3) = 25$

Simplifying this:

$3x + 5 - 2x + 3 = 25$

$x + 8 = 25 \quad \Rightarrow \quad x = 17$

Case 3: $3x + 5 < 0$ and $2x - 3 \geq 0$

In this case, $3x + 5$ is negative, and $2x - 3$ is positive. The equation becomes:

$-(3x + 5) + (2x - 3) = 25$

Simplifying this:

$-3x - 5 + 2x - 3 = 25$

$-x - 8 = 25 \quad \Rightarrow \quad -x = 33 \quad \Rightarrow \quad x = -33$

Case 4: $3x + 5 < 0$ and $2x - 3 < 0$

Here, both expressions inside the absolute values are negative. The equation becomes:

$-(3x + 5) - (2x - 3) = 25$

Simplifying this:

$-3x - 5 - 2x + 3 = 25$

$-5x - 2 = 25 \quad \Rightarrow \quad -5x = 27 \quad \Rightarrow \quad x = -\frac{27}{5} = -5.4$

Summary of Solutions:

The possible solutions for $x$ are:

• $x = 4.6$
• $x = 17$
• $x = -33$
• $x = -5.4$

Finding the Sum of the Highest and Lowest Values:

• The highest value of $x$ is $17$.
• The lowest value of $x$ is $-33$.

Thus, the sum of the highest and lowest values is:

$17 + (-33) = -16$

Final Answer:

The sum of the highest and lowest values is $-16$.

Do you want more details on any step, or have any questions? Here are some related questions to explore:

1. How do absolute value equations affect solutions in different cases?
2. What is the importance of considering different sign changes in absolute value functions?
3. How can graphing the equation help visualize the solution?
4. What are the methods to solve absolute value inequalities?
5. How can absolute values be applied in real-world problems?

Tip: When solving absolute value equations, always break the equation into cases to handle the positive and negative parts separately.