the value of x satisfying |x-4| + |x-9| = 5 is

To solve the equation \(|x-4| + |x-9| = 5\), we need to consider the points where the expressions inside the absolute values change sign. These points are \(x = 4\) and \(x = 9\). We can break down the problem into three intervals based on these points:

1. \(x < 4\)
2. \(4 \leq x \leq 9\)
3. \(x > 9\)

Let's analyze each interval:

### 1. Interval \(x < 4\)
In this interval, both \(x-4\) and \(x-9\) are negative. Thus, we have:
\[ |x-4| = 4-x \]
\[ |x-9| = 9-x \]
So the equation becomes:
\[ (4-x) + (9-x) = 5 \]
\[ 13 - 2x = 5 \]
\[ 13 - 5 = 2x \]
\[ 8 = 2x \]
\[ x = 4 \]
However, \(x = 4\) does not belong to the interval \(x < 4\). So, there is no solution in this interval.

### 2. Interval \(4 \leq x \leq 9\)
In this interval, \(x-4\) is non-negative and \(x-9\) is negative. Thus, we have:
\[ |x-4| = x-4 \]
\[ |x-9| = 9-x \]
So the equation becomes:
\[ (x-4) + (9-x) = 5 \]
\[ x - 4 + 9 - x = 5 \]
\[ 5 = 5 \]
This is always true for any \(x\) in the interval \([4, 9]\). Therefore, any \(x\) in this interval satisfies the equation.

### 3. Interval \(x > 9\)
In this interval, both \(x-4\) and \(x-9\) are non-negative. Thus, we have:
\[ |x-4| = x-4 \]
\[ |x-9| = x-9 \]
So the equation becomes:
\[ (x-4) + (x-9) = 5 \]
\[ 2x - 13 = 5 \]
\[ 2x = 18 \]
\[ x = 9 \]
However, \(x = 9\) does not belong to the interval \(x > 9\). So, there is no solution in this interval.

Therefore, the solution to the equation \(|x-4| + |x-9| = 5\) is \(x \in [4, 9]\).

Would you like more details or have any questions?

Here are 5 questions you might want to ask:
1. How do we solve absolute value equations?
2. Can you explain more about how to handle different intervals for absolute value equations?
3. What are some example problems involving absolute value functions?
4. How do absolute value inequalities work?
5. Can you show how to graph the function \(|x-4| + |x-9|\)?

**Tip:** When solving absolute value equations, always consider the points where the expressions inside the absolute values change sign, and analyze each resulting interval separately.

Learn how to solve the absolute value equation |x-4| + |x-9| = 5 with a detailed step-by-step solution. This problem involves analyzing different intervals and understanding how absolute values affect equations. Suitable for Grades 9-12.

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