Understanding Absolute Value: Solving |x - 3| = 7

Understanding Absolute Value: Solving |x - 3| = 7

When you first encounter an equation like |x - 3| = 7, it might seem a bit like a puzzle—an intriguing challenge waiting to be solved. First off, let’s break down what absolute value means in straightforward terms. You know, it’s all about distance from zero on the number line. It’s the non-negative value that tells us how far a number is from zero, regardless of direction. So, if you see |x - 3|, just think of it as the distance between x and 3.

A Look at the Equation

This leads us directly to the equation in question: |x - 3| = 7. What are we really saying here? We’re suggesting that the distance between x and 3 is 7. But wait—this can happen in two different ways! We can either be 7 units to the right (positive distance) or 7 units to the left (negative distance) of 3. Hence, we can split this into two equations:

1. x - 3 = 7
2. x - 3 = -7

Solving Step by Step

Let’s tackle the first equation. To isolate x, all we need to do is add 3 to both sides:

• x - 3 = 7
• x = 7 + 3
• x = 10

Bam! We have one solution: x = 10.

Now, let's switch gears and look at the second equation:

• x - 3 = -7
• x = -7 + 3
• x = -4

And there we go! Our second solution pops right off the page: x = -4.

The Final Words

So, if you think about it, solving absolute value equations is all about understanding that they can have dual solutions. In our case, the pair of answers from |x - 3| = 7 are x = 10 and x = -4. Pretty nifty, right? Knowing how to work with absolute values not only helps you ace problems in your Ontario Mathematics Proficiency Test, but it’s also a crucial building block for more advanced math topics.

Why This Matters

Being able to grasp these concepts doesn’t just help you pass an exam; it builds a foundation for logical reasoning and critical thinking—skills that stretch way beyond the classroom walls!

In Conclusion

So next time you face an absolute value problem, remember: It’s really just about finding distance. And, hey, with a bit of practice, you’ll approach these questions with confidence. The more you tackle, the more intuitive they become. You got this!