Question

question
solve the equation for all values of x.
|4x - 11| + 6 = 3x
answer
attempt 1 out of 2

Explanation:

Step1: Isolate the absolute value

Subtract 6 from both sides.
$$|4x - 1| = 3x - 6$$

Step2: Define domain for right-hand side

Absolute value is non-negative, so $3x - 6 \geq 0$.
$$3x \geq 6 \implies x \geq 2$$

Step3: Split into two cases

Case 1: $4x - 1 = 3x - 6$
Solve for $x$:
$$4x - 3x = -6 + 1 \implies x = -5$$
Case 2: $4x - 1 = -(3x - 6)$
Solve for $x$:
$$4x - 1 = -3x + 6 \implies 4x + 3x = 6 + 1 \implies 7x = 7 \implies x = 1$$

Step4: Validate solutions

Check $x=-5$: $-5 < 2$, violates domain. Reject.
Check $x=1$: $1 < 2$, violates domain. Reject.

Answer:

No real solutions exist.