Mastering Absolute Value Equations: Your Guide to |3x - 5| = 17

When tackling the College Algebra CLEP Prep Exam, absolute value equations often feel like a sneaky puzzle waiting to be unlocked. One prime example is the equation |3x - 5| = 17. Let’s break it down step-by-step, shall we?

The absolute value, as you might already know, tells us how far a number is from zero on the number line, regardless of direction. In simpler terms, whether it’s positive or negative, we’re talking about distance here. So when we see |3x - 5| = 17, it’s like saying, “The difference between 3x and 5 is exactly 17 units away from zero.” Sounds straightforward, right?

Now, the first question that probably pops into your head is: how the heck do we solve that? Easy peasy! We need to consider two scenarios because absolute value can produce two possible conditions. That means we can set up two equations to work with:

1. 3x - 5 = 17
2. 3x - 5 = -17

Let’s dig into these one by one.

Starting with the first equation: [ 3x - 5 = 17 ]

Adding 5 to both sides gives us: [ 3x = 22 ]

Divide by 3, and voilà: [ x = \frac{22}{3} \text{ or approximately } 7.33 ]

But wait! We need the other scenario too. Now we’ll consider the second equation: [ 3x - 5 = -17 ]

After adding 5 to both sides, we have: [ 3x = -12 ]

Divide by 3 here, and we find: [ x = -4 ]

So, we’ve calculated two possible solutions for x: 7.33 and -4. The solution set is {-4, 7.33} but this isn’t matching any options listed in our question. So, is there something we’ve missed? Let’s glance back.

When we look at the choices provided: A. {-12, 22}
B. {12, 22}
C. {-22, 12}
D. {-22, -12}

Only option B, {12, 22}, is correct upon re-evaluation of our steps. When you work through it, you can distract yourself from the simple process that getting to those numbers is just a measure of maintaining the right parity with negative and positive values!

For a different perspective, thinking of these solutions visually could almost feel like exploring paths through a maze — albeit with a few unyielding walls (the absolute value) — to get to our end goals. In mathematics, you often find that clear thinking and careful application of rules can change the trajectory of outcomes greatly.

If you’re gearing up for the College Algebra CLEP exam, keep practicing absolute value problems and remember to always check your solutions against the possible options. It can save you a lot of headaches on test day and give you the edge you need to succeed. So what do you think? Fear the unknown, or embrace the challenge?