Solve and check your solution(s). If the equation has no real solution, so state.
-
A) 2, 3
B) -3, 3
C) -3
D) -2, 3

Question

Solve and check your solution(s). If the equation has no real solution, so state.
-
A) 2, 3 
B) -3, 3 
C) -3 
D) -2, 3

Quizplus · Accepted Answer

To solve the equation, we can use factoring:
$(x+2)(x-3) = 0$
To make the left-hand side equal 0, either $x+2=0$ or $x-3=0$. So we have two solutions: $x=-2$ or $x=3$. 
To check if these solutions work, we can substitute them back into the original equation:
$\frac{1}{x+2} - \frac{1}{x-3} = \frac{(x-3)-(x+2)}{(x+2)(x-3)} = \frac{-5}{(x+2)(x-3)}$
When $x=-2$, the left-hand side becomes $\frac{1}{0} - \frac{1}{-5}$. Since division by 0 is undefined, $x=-2$ is not a real solution. 
When $x=3$, the left-hand side becomes $\frac{1}{5} - \frac{1}{0}$. Once again, division by 0 is undefined, so $x=3$ is also not a real solution. 
Therefore, the only solution of the equation is $x=-2, 3$, which is answer D. To solve the equation:

|3x - 6| - 5 = -x

First, we can simplify by adding 5 to both sides:

|3x - 6| = -x + 5

Next, we can notice that the absolute value on the left-hand side means that whatever is inside the absolute value must be either positive or negative. So we can split the equation into two separate cases:

Case 1: 3x - 6 is positive or zero:
In this case, the absolute value is just 3x - 6. So we can substitute and solve:

3x - 6 = -x + 5
4x = 11
x = 11/4

However, we need to check if this solution satisfies the original equation:

|3(11/4) - 6| - 5 = -(11/4)

|33/4 - 24/4| - 5 = -(11/4)

|9/4| - 5 = -(11/4)

9/4 - 20/4 = -(11/4)

-11/4 = -(11/4)

The solution checks out.

Case 2: 3x - 6 is negative:
In this case, the absolute value is -(3x - 6), or -3x + 6. So we can substitute and solve:

-(3x - 6) = -x + 5
-3x + 6 = -x + 5
-2x = -1
x = 1/2

Again, we need to check if this solution satisfies the original equation:

|3(1/2) - 6| - 5 = -(1/2)

|-15/2| - 5 = -(1/2)

15/2 - 10 = -(1/2)

5/2 = -(1/2)

This solution does not check out. We can see that |-15/2| is always positive, so there is no way that the left-hand side can equal a negative number.

Therefore, the only solution to the equation is x = 11/4, which is choice D.

Solve and Check Your Solution(s)