Question

Compare and Contrast: Two equations are listed below. Solve each equation and compare the solutions. Choose the statement that is true about both solutions.


Equation 1
|5x − 6| = −41
Equation 2
|7x + 13| = 27

a) Equation 1 has more solutions than equation 2.
b) Equation 1 and Equation 2 have the same number of solutions.
c) Equation 2 has more solutions than Equation 1.
d) The number of solutions cannot be determined.

Answer 1

Okay, so the absolute value of x is equal to the number of spaces that x is away from 0 on a number line. To demonstrate:

The absolute value of 3 is 3. We need to move 3 spaces to the left to reach 0.
The absolute value of -3 is ALSO 3, as we can still reach 0 in 3 spaces, we just need to move right this time.

Now that we've covered that base, let's look at the questions one at a time.

|5x − 6| = −41

Until you reach it's most simplified form, this equation is exactly the same as it would be if there was no absolute value.

|5x − 6| = −41
|5x| = -35
|x| = -7

So, what number is -7 spaces away from 0 on a number line?

Your proper response should be "WTF are you talking about, Salivanth, how do you go -7 spaces on a number line? You're silly."

Well, you're right! This equation in fact has NO SOLUTIONS.

Second one:

|7x + 13| = 27

Again, we simplify to it's simplest form before we need to worry about absolute value at all.

|7x| = 14.
|x| = 2.

So the absolute value of x is 2. That means that x can equal 2 or -2, because -2 and 2 are both 2 spaces away from 0.

So, which statement is true here? By process of elimination, going through each statement in turn, our answer is C.