Answer :
We start with the equation
[tex]$$
4|x - 5| + 3 = 15.
$$[/tex]
Step 1. Subtract 3 from both sides:
[tex]$$
4|x - 5| = 15 - 3 = 12.
$$[/tex]
Step 2. Divide both sides by 4:
[tex]$$
|x - 5| = \frac{12}{4} = 3.
$$[/tex]
Step 3. Solve the absolute value equation. Recall that if [tex]$$|z| = 3,$$[/tex] then [tex]$$z = 3$$[/tex] or [tex]$$z = -3.$$[/tex] Here, let [tex]$$z = x - 5.$$[/tex] This gives two equations:
[tex]$$
x - 5 = 3 \quad \text{or} \quad x - 5 = -3.
$$[/tex]
Step 4. Solve each equation separately.
For [tex]$$x - 5 = 3:$$[/tex]
[tex]$$
x = 3 + 5 = 8.
$$[/tex]
For [tex]$$x - 5 = -3:$$[/tex]
[tex]$$
x = -3 + 5 = 2.
$$[/tex]
Final Answer: The values of [tex]$$x$$[/tex] that satisfy the equation are [tex]$$x = 2$$[/tex] and [tex]$$x = 8.$$[/tex]
[tex]$$
4|x - 5| + 3 = 15.
$$[/tex]
Step 1. Subtract 3 from both sides:
[tex]$$
4|x - 5| = 15 - 3 = 12.
$$[/tex]
Step 2. Divide both sides by 4:
[tex]$$
|x - 5| = \frac{12}{4} = 3.
$$[/tex]
Step 3. Solve the absolute value equation. Recall that if [tex]$$|z| = 3,$$[/tex] then [tex]$$z = 3$$[/tex] or [tex]$$z = -3.$$[/tex] Here, let [tex]$$z = x - 5.$$[/tex] This gives two equations:
[tex]$$
x - 5 = 3 \quad \text{or} \quad x - 5 = -3.
$$[/tex]
Step 4. Solve each equation separately.
For [tex]$$x - 5 = 3:$$[/tex]
[tex]$$
x = 3 + 5 = 8.
$$[/tex]
For [tex]$$x - 5 = -3:$$[/tex]
[tex]$$
x = -3 + 5 = 2.
$$[/tex]
Final Answer: The values of [tex]$$x$$[/tex] that satisfy the equation are [tex]$$x = 2$$[/tex] and [tex]$$x = 8.$$[/tex]
