Answer :
We start with the equation
[tex]$$
4|x-5| + 3 = 15.
$$[/tex]
Step 1: Isolate the absolute term
Subtract 3 from both sides to get:
[tex]$$
4|x-5| = 15 - 3 = 12.
$$[/tex]
Step 2: Solve for the absolute value
Divide both sides by 4:
[tex]$$
|x-5| = \frac{12}{4} = 3.
$$[/tex]
Step 3: Solve the absolute value equation
The equation [tex]$|x-5| = 3$[/tex] has two cases:
1. Case 1: When the expression inside the absolute value is positive:
[tex]$$
x-5 = 3 \quad \Longrightarrow \quad x = 5 + 3 = 8.
$$[/tex]
2. Case 2: When the expression inside the absolute value is negative:
[tex]$$
x-5 = -3 \quad \Longrightarrow \quad x = 5 - 3 = 2.
$$[/tex]
Thus, the solutions are [tex]$x = 2$[/tex] and [tex]$x = 8$[/tex].
Final Answer: [tex]$x = 2 \text{ or } x = 8$[/tex].
[tex]$$
4|x-5| + 3 = 15.
$$[/tex]
Step 1: Isolate the absolute term
Subtract 3 from both sides to get:
[tex]$$
4|x-5| = 15 - 3 = 12.
$$[/tex]
Step 2: Solve for the absolute value
Divide both sides by 4:
[tex]$$
|x-5| = \frac{12}{4} = 3.
$$[/tex]
Step 3: Solve the absolute value equation
The equation [tex]$|x-5| = 3$[/tex] has two cases:
1. Case 1: When the expression inside the absolute value is positive:
[tex]$$
x-5 = 3 \quad \Longrightarrow \quad x = 5 + 3 = 8.
$$[/tex]
2. Case 2: When the expression inside the absolute value is negative:
[tex]$$
x-5 = -3 \quad \Longrightarrow \quad x = 5 - 3 = 2.
$$[/tex]
Thus, the solutions are [tex]$x = 2$[/tex] and [tex]$x = 8$[/tex].
Final Answer: [tex]$x = 2 \text{ or } x = 8$[/tex].
