How do you solve #|x + 3| = (x-2)#?

Question

Kennedy Adams · Accepted Answer

No solution

Since absolute values are always nonnegative, #|x+3|=x-2 geq0 Rightarrow x geq 2#, which means that #x+3# is nonnegative. So, the equation becomes (by simply removing the absolute value sign) #Rightarrow x+3=x-2# By subtracting #x# from both sides, #Rightarrow 3=-2#, which is false. Hence, there is no solution. I hope that this was clear.

Isaiah Anthony · Answer

No real solution.

We have #abs(x+3)=x+3-5# or assuming #x ne -3# #1=(x+3)/abs(x+3)-5/abs(x+3)# so there are two possibilities #{(1=1-5/abs(x+3)),(1=-1-5/abs(x+3)):}# and in both cases there is not a real solution for the equations.

Genesis Allen · Answer

undefined To solve the equation $|x + 3| = x - 2$, we need to consider two cases: when $x + 3$ is positive and when $x + 3$ is negative.

Case 1: $x + 3 \geq 0$:
In this case, the absolute value of $x + 3$ is equal to $x + 3$ itself. So, we have:
$$x + 3 = x - 2$$
$$3 = -2$$

This equation has no solution because it leads to a contradiction.

Case 2: $x + 3 < 0$:
In this case, the absolute value of $x + 3$ is equal to $-(x + 3)$. So, we have:
$$-(x + 3) = x - 2$$
$$-x - 3 = x - 2$$
$$-3 = 2x - 2$$
$$2x = -1$$
$$x = -\frac{1}{2}$$

Therefore, the solution to the equation is $x = -\frac{1}{2}$.