Question

question
solve the equation for all values of x.
$|5x - 4| + 5 = 4x$

Explanation:

Step1: Isolate the absolute value

Subtract 5 from both sides.
$$|5x - 4| = 4x - 5$$

Step2: Define domain for RHS

Right-hand side must be non-negative.
$$4x - 5 \geq 0 \implies x \geq \frac{5}{4}$$

Step3: Case 1: Inside absolute value non-negative

Set $5x - 4 \geq 0$, solve linear equation.
$$5x - 4 = 4x - 5$$
$$5x - 4x = -5 + 4$$
$$x = -1$$
This value fails $x \geq \frac{5}{4}$, so discard.

Step4: Case 2: Inside absolute value negative

Set $5x - 4 < 0$, solve linear equation.
$$-(5x - 4) = 4x - 5$$
$$-5x + 4 = 4x - 5$$
$$-5x - 4x = -5 - 4$$
$$-9x = -9$$
$$x = 1$$
This value fails $x \geq \frac{5}{4}$, so discard.

Step5: Verify no valid solutions

Both solutions do not satisfy the domain constraint.

Answer:

No real solutions exist.