Question

solve \\(|x - 3| = 4\\).

select the correct answer

\\(x = 7\\)

\\(x = -1\\)

both \\(x = 7\\) and \\(x = -1\\)

no solution

Explanation:

Set up the absolute value cases

An absolute value equation of the form \(|u| = c\) (where \(c \ge 0\)) splits into two separate linear equations:

$$ u = c \quad \text{or} \quad u = -c $$

For the given equation \(|x - 3| = 4\), we set up the two cases:

$$ x - 3 = 4 \quad \text{or} \quad x - 3 = -4 $$

Solve the first case

To solve the first equation, add \(3\) to both sides:

$$ x - 3 = 4 \implies x = 4 + 3 \implies x = 7 $$

Solve the second case

To solve the second equation, add \(3\) to both sides:

$$ x - 3 = -4 \implies x = -4 + 3 \implies x = -1 $$

Verify the solutions

Substitute both values back into the original equation to check validity:

• For \(x = 7\): \(|7 - 3| = |4| = 4\) (True)
• For \(x = -1\): \(|-1 - 3| = |-4| = 4\) (True)

Both values are valid solutions.

Answer:

• x = 7
• x = -1
• Both x = 7 and x = -1 (Correct answer)
• No Solution