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Have you ever encountered a math problem that looks like a jumbled mess of fractions, variables, and that strange ‘greater than’ or ‘less than’ symbol? Don't worry, you're not alone! Many people find inequalities with fractions and variables intimidating at first. But the truth is, they're like any other math problem – with a little patience and the right approach, you can solve them with confidence.
Let's break down the process step-by-step, using an example to guide us:
Imagine you're faced with this inequality:
2/3 > -4y - 8 1/3
Step 1: Out with the Mixed Numbers!
Mixed numbers like 8 1/3 can be a bit clunky to work with. Let's convert it into a simpler fraction. Remember, 8 1/3 means 8 + 1/3. To combine them, we do this:
- Denominator stays the same: Our denominator is 3.
- New numerator: (8 x 3) + 1 = 25
So, 8 1/3 is the same as 25/3. Now our inequality looks like this:
2/3 > -4y - 25/3
Step 2: Fractions Be Gone!
Fractions can be pesky, so let's get rid of them! Find the least common denominator (in this case, it's 3) and multiply both sides of the inequality by it.
(3) * (2/3) > (3) * (-4y - 25/3)
This simplifies to:
2 > -12y - 25
Step 3: Variables on One Side, Constants on the Other
Our goal is to get that 'y' by itself. To do this, we need to move the constant terms to the other side of the inequality. We can add 25 to both sides:
2 + 25 > -12y - 25 + 25
This gives us:
27 > -12y
Step 4: Isolate the Variable
Almost there! To get 'y' completely alone, we divide both sides by -12. But remember, when you divide or multiply both sides of an inequality by a negative number, you must flip the inequality sign!
27 / -12 < -12y / -12
This simplifies to:
-9/4 < y
Step 5: Rewrite and Celebrate!
We've solved for 'y'! It's common to write the variable on the left side, so we can rewrite our solution as:
y > -9/4
What does this mean?
Our solution tells us that any value of 'y' that is greater than -9/4 will make the original inequality true.
Key Takeaways
- Don't be intimidated by fractions or variables – they're just pieces of the puzzle!
- Break down the problem into smaller, manageable steps.
- Remember the golden rule: When multiplying or dividing by a negative number, flip the inequality sign!
With practice, you'll be solving inequalities with fractions and variables like a pro in no time! And remember, math is a journey of learning and growth – embrace the challenge and celebrate your successes along the way!
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