Question

1. solve: $|x+3|=4x$

Explanation:

Step1: Define absolute value case 1

When $x+3 \geq 0$ (i.e., $x \geq -3$), $|x+3|=x+3$. Set equal to $4x$:
$x+3=4x$

Step2: Solve case 1 equation

Subtract $x$ from both sides:
$3=3x$
Divide by 3:
$x=1$
Verify: $1 \geq -3$, and $|1+3|=4=4(1)$, valid.

Step3: Define absolute value case 2

When $x+3 < 0$ (i.e., $x < -3$), $|x+3|=-(x+3)$. Set equal to $4x$:
$-(x+3)=4x$

Step4: Solve case 2 equation

Expand left side:
$-x-3=4x$
Add $x$ to both sides:
$-3=5x$
Divide by 5:
$x=-\frac{3}{5}$
Verify: $-\frac{3}{5}=-0.6$ which is not $< -3$, invalid.

Step5: Discard invalid solution

Only $x=1$ satisfies the original equation and domain.

Answer:

$x=1$