Question

question
solve the equation for all values of x.
\\(|3x + 1| + 5 = x|
answer attempt 1 out of 2
\\(\odot\\) additional solution \\(\odot\\) no solution
\\(x =\\)

Explanation:

Step1: Isolate the absolute value

Subtract 5 from both sides: $|3x + 1| = x - 5$

Step2: Analyze the absolute value property

The absolute value $|A| \geq 0$ for any real number $A$. So $x - 5 \geq 0 \implies x \geq 5$

Step3: Case 1: $3x + 1 \geq 0 \implies x \geq -\frac{1}{3}$

Then $|3x + 1| = 3x + 1$, so the equation becomes $3x + 1 = x - 5 \implies 2x = -6 \implies x = -3$
But $x = -3$ does not satisfy $x \geq 5$ (from Step 2), so discard.

Step4: Case 2: $3x + 1 < 0 \implies x < -\frac{1}{3}$

Then $|3x + 1| = - (3x + 1) = -3x - 1$, so the equation becomes $-3x - 1 = x - 5 \implies -4x = -4 \implies x = 1$
But $x = 1$ does not satisfy $x \geq 5$ (from Step 2) and also $x = 1$ does not satisfy $x < -\frac{1}{3}$ (from Case 2), so discard.

Step5: Conclusion

No values of $x$ satisfy the equation.

Answer:

No Solution