What is the sum of the solutions to the given equation?
Understand the Problem
The question is asking for the sum of the solutions to an absolute value equation. We will solve the equation |x - 9| + 45 = 63 to find the values of x and then calculate their sum.
Answer
The sum of the solutions is $18$.
Answer for screen readers
The sum of the solutions is $18$.
Steps to Solve
- Isolate the absolute value expression
Subtract 45 from both sides of the equation:
$$ |x - 9| + 45 - 45 = 63 - 45 $$
This simplifies to:
$$ |x - 9| = 18 $$
- Set up two cases for the absolute value
The absolute value equation $|x - 9| = 18$ can be rewritten as two separate equations:
$$ x - 9 = 18 $$ and $$ x - 9 = -18 $$
- Solve the first case
For the first equation:
$$ x - 9 = 18 $$
Add 9 to both sides:
$$ x = 18 + 9 $$
Thus,
$$ x = 27 $$
- Solve the second case
For the second equation:
$$ x - 9 = -18 $$
Add 9 to both sides:
$$ x = -18 + 9 $$
Thus,
$$ x = -9 $$
- Calculate the sum of the solutions
Now that we have the solutions, $x = 27$ and $x = -9$, we can find the sum:
$$ 27 + (-9) = 18 $$
The sum of the solutions is $18$.
More Information
The absolute value represents the distance from zero on the number line, so solving the absolute value equation often results in two possible scenarios. The solutions can be visualized as the points where the function intersects with a horizontal line.
Tips
- Forgetting to consider both cases of the absolute value can lead to missing solutions.
- Failing to isolate the absolute value before splitting into cases can cause errors in calculations.
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