Question Solve the absolute value equation $2.5|x+4|=0.5|x+4|+10$ and classify the solution(s).

Studdy Solution

STEP 1

Assumptions
1. We are given the equation $2.5|x+4| = 0.5|x+4| + 10$ .
2. We need to solve for $x$ .
3. The solution set will be classified into number types: counting numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers.

STEP 2

To solve the equation, we will first simplify it by moving terms involving $|x+4|$ to one side of the equation.
$2.5|x+4| - 0.5|x+4| = 10$

STEP 3

Combine like terms on the left-hand side of the equation.
$2.0|x+4| = 10$

STEP 4

Divide both sides of the equation by $2.0$ to isolate $|x+4|$ .
$|x+4| = \frac{10}{2.0}$

STEP 5

Calculate the right-hand side of the equation.
$|x+4| = 5$

STEP 6

The absolute value equation $|x+4| = 5$ has two possible solutions: one where the quantity inside the absolute value is positive and equal to 5, and one where it is negative and equal to -5.

STEP 7

First, let's consider the case where $x+4$ is positive.
$x+4 = 5$

STEP 8

Subtract 4 from both sides of the equation to solve for $x$ .
$x = 5 - 4$

STEP 9

Calculate the value of $x$ for this case.
$x = 1$

STEP 10

Now, let's consider the case where $x+4$ is negative.
$x+4 = -5$

STEP 11

Subtract 4 from both sides of the equation to solve for $x$ .
$x = -5 - 4$

STEP 12

Calculate the value of $x$ for this case.
$x = -9$

STEP 13

We have found two solutions for $x$ : $x = 1$ and $x = -9$ .

STEP 14

Classify the solutions according to the given number types.
1 is a counting number, whole number, integer, rational number, and real number. -9 is an integer, rational number, and real number.

STEP 15

We conclude that the solutions are:
$x = 1$ which is a counting number, whole number, integer, rational number, and real number. $x = -9$ which is an integer, rational number, and real number.

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