Question Find the value of $a$ if the solutions to the equation $|x+a| = 4$ are $-2$ and $6$ .

Studdy Solution

STEP 1

1. The equation $|x+a|=4$ represents an absolute value equation, which can have two solutions, one positive and one negative.
2. The solutions provided, -2 and 6, correspond to the two cases of the absolute value: $x+a=4$ and $x+a=-4$ .
3. The value of $a$ can be found by solving these two equations with the given solutions.

STEP 2

1. Set up the two possible equations based on the absolute value.
2. Solve the first equation using the solution $x=-2$ .
3. Solve the second equation using the solution $x=6$ .
4. Verify that the value of $a$ satisfies both solutions.

STEP 3

Set up the two possible equations that come from the absolute value equation $|x+a|=4$ .
$x + a = 4 \quad \text{and} \quad x + a = -4$

STEP 4

Solve the first equation $x + a = 4$ using the solution $x = -2$ .
$-2 + a = 4$

STEP 5

Isolate the variable $a$ by adding 2 to both sides of the equation.
$a = 4 + 2$

STEP 6

Calculate the value of $a$ .
$a = 6$

STEP 7

Solve the second equation $x + a = -4$ using the solution $x = 6$ .
$6 + a = -4$

STEP 8

Isolate the variable $a$ by subtracting 6 from both sides of the equation.
$a = -4 - 6$

STEP 9

Calculate the value of $a$ .
$a = -10$

STEP 10

Verify that the value of $a$ satisfies both solutions. Since we have two different values for $a$ , we need to check which one is consistent with both solutions given for the equation $|x+a|=4$ .
Check $a=6$ with $x=-2$ : $|-2+6| = |4| = 4$ This is true.
Check $a=-10$ with $x=6$ : $|6-10| = |-4| = 4$ This is also true.
Therefore, both values of $a$ satisfy the given solutions for the absolute value equation.
The value of $a$ that satisfies both solutions of the equation $|x+a|=4$ is $a=6$ and $a=-10$ .

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