Solving Inequalities: A Step-by-Step Guide

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Hey guys! Today, we're diving into the world of inequalities and tackling a set of problems that might seem a bit daunting at first. But don't worry, we'll break them down step by step, so you'll be solving inequalities like a pro in no time! We're going to be focusing on solving inequalities where x is a real number. So, let's get started!

Understanding Inequalities

Before we jump into the solutions, let's quickly recap what inequalities are. Unlike equations, which show equality between two expressions, inequalities show a relationship where one expression is greater than, less than, greater than or equal to, or less than or equal to another expression. The symbols we use are:

  • < (less than)

  • (greater than)

  • ≤ (less than or equal to)
  • ≥ (greater than or equal to)

Think of it like this: imagine a see-saw. With an equation, the see-saw is perfectly balanced. With an inequality, one side is higher or lower than the other. Our goal is to find the range of values for x that keep the see-saw tilted in the correct direction.

The Golden Rules of Solving Inequalities

Solving inequalities is very similar to solving equations, but there's one crucial difference. Here are the golden rules to keep in mind:

  1. Isolate the variable: Just like with equations, our main aim is to get x by itself on one side of the inequality.
  2. Perform the same operations on both sides: Whatever you do to one side of the inequality, you must do to the other to maintain the balance (or rather, the imbalance!). You can add, subtract, multiply, or divide both sides by the same number.
  3. The Flip Rule: This is the big one! If you multiply or divide both sides of the inequality by a negative number, you must flip the inequality sign. This is because multiplying or dividing by a negative number reverses the order of the numbers on the number line. For instance, if 2 < 3, then -2 > -3. This rule is crucial for getting the correct answer.
  4. Simplify: Always simplify your answer as much as possible. This makes it easier to understand and interpret.

Now that we've got the basics covered, let's dive into the problems!

Solving the Inequalities

We have six inequalities to solve, each with its own little twist. We'll go through them one by one, showing each step clearly. Remember, the key is to isolate x while following those golden rules!

a) x + 5/4 < 2 3/4

Our first inequality is x + 5/4 < 2 3/4. The goal here is to get x by itself on the left side. So, we need to get rid of the 5/4. How do we do that? We subtract 5/4 from both sides. Remember, whatever we do to one side, we do to the other!

  • x + 5/4 < 2 3/4
  • x + 5/4 - 5/4 < 2 3/4 - 5/4

Now, let's simplify. First, we need to convert the mixed number 2 3/4 into an improper fraction. To do this, we multiply the whole number (2) by the denominator (4) and add the numerator (3), then put the result over the original denominator:

  • 2 3/4 = (2 * 4 + 3) / 4 = 11/4

So, our inequality now looks like this:

  • x < 11/4 - 5/4

Now we can subtract the fractions since they have the same denominator:

  • x < 6/4

Finally, we can simplify the fraction 6/4 by dividing both the numerator and denominator by their greatest common divisor, which is 2:

  • x < 3/2

So, the solution to the first inequality is x < 3/2. This means that any real number less than 3/2 will satisfy the inequality. We can also write this in interval notation as (-∞, 3/2).

b) x - 3/5 > 1 1/5

Next up, we have x - 3/5 > 1 1/5. Again, our mission is to isolate x. This time, we have -3/5 on the left side, so we need to add 3/5 to both sides:

  • x - 3/5 > 1 1/5
  • x - 3/5 + 3/5 > 1 1/5 + 3/5

Let's convert the mixed number 1 1/5 into an improper fraction:

  • 1 1/5 = (1 * 5 + 1) / 5 = 6/5

Now our inequality is:

  • x > 6/5 + 3/5

Add the fractions:

  • x > 9/5

So, the solution to the second inequality is x > 9/5. Any real number greater than 9/5 will satisfy the inequality. In interval notation, this is (9/5, ∞).

c) x + 2/3 ≥ 2 1/3

Moving on, we have x + 2/3 ≥ 2 1/3. This time, we have a “greater than or equal to” sign, but the process is exactly the same. We need to isolate x, so we subtract 2/3 from both sides:

  • x + 2/3 ≥ 2 1/3
  • x + 2/3 - 2/3 ≥ 2 1/3 - 2/3

Convert the mixed number 2 1/3 to an improper fraction:

  • 2 1/3 = (2 * 3 + 1) / 3 = 7/3

Now we have:

  • x ≥ 7/3 - 2/3

Subtract the fractions:

  • x ≥ 5/3

So, the solution to this inequality is x ≥ 5/3. This means that any real number greater than or equal to 5/3 will satisfy the inequality. In interval notation, this is [5/3, ∞).

d) 3 1/6 - x ≤ 15/6

Now, this one's a little different! We have 3 1/6 - x ≤ 15/6. Notice that x is being subtracted. Don't panic! We can still solve this. First, let's convert the mixed number 3 1/6 to an improper fraction:

  • 3 1/6 = (3 * 6 + 1) / 6 = 19/6

So our inequality is:

  • 19/6 - x ≤ 15/6

We want to isolate x, so let's subtract 19/6 from both sides:

  • 19/6 - x - 19/6 ≤ 15/6 - 19/6
  • -x ≤ -4/6

Now, here's the crucial step! We have -x on the left side, but we want x. To get rid of the negative sign, we need to multiply both sides by -1. And remember the Flip Rule! When we multiply by a negative number, we have to flip the inequality sign:

  • (-1) * (-x) ≥ (-1) * (-4/6)
  • x ≥ 4/6

We can simplify the fraction 4/6 by dividing both the numerator and denominator by 2:

  • x ≥ 2/3

So, the solution to this inequality is x ≥ 2/3. In interval notation, this is [2/3, ∞).

e) 4 3/8 - x > 15/8

This inequality is similar to the previous one. We have 4 3/8 - x > 15/8. Let's start by converting the mixed number to an improper fraction:

  • 4 3/8 = (4 * 8 + 3) / 8 = 35/8

Now we have:

  • 35/8 - x > 15/8

Subtract 35/8 from both sides:

  • 35/8 - x - 35/8 > 15/8 - 35/8
  • -x > -20/8

Multiply both sides by -1 and flip the inequality sign:

  • (-1) * (-x) < (-1) * (-20/8)
  • x < 20/8

Simplify the fraction 20/8 by dividing both numerator and denominator by 4:

  • x < 5/2

So, the solution to this inequality is x < 5/2. In interval notation, this is (-∞, 5/2).

f) 5 5/9 - x < 20/9

Last but not least, we have 5 5/9 - x < 20/9. Let's convert the mixed number:

  • 5 5/9 = (5 * 9 + 5) / 9 = 50/9

Our inequality is now:

  • 50/9 - x < 20/9

Subtract 50/9 from both sides:

  • 50/9 - x - 50/9 < 20/9 - 50/9
  • -x < -30/9

Multiply both sides by -1 and flip the inequality sign:

  • (-1) * (-x) > (-1) * (-30/9)
  • x > 30/9

Simplify the fraction 30/9 by dividing both numerator and denominator by 3:

  • x > 10/3

So, the solution to the final inequality is x > 10/3. In interval notation, this is (10/3, ∞).

Conclusion

And there you have it! We've solved all six inequalities. Remember, the key to solving inequalities is to isolate x while paying close attention to the Flip Rule. Don't forget, practice makes perfect, so try solving some more inequalities on your own. You've got this! I hope this guide helped you guys understand inequalities a little better. Keep practicing, and you'll be a math whiz in no time!