Solving Inequalities: Step-by-Step Guide

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Hey guys! Today, we're diving into the world of inequalities. Don't worry, it's not as scary as it sounds. We'll break down each problem step by step so you can totally nail it. Let's get started!

Understanding Inequalities

Before we jump into solving, let's quickly recap what inequalities are all about. Unlike equations that use an equals sign (=), inequalities use symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). The goal is to find the range of values that make the inequality true. Think of it like finding all the possible numbers that fit the condition.

When you're dealing with inequalities, there's one golden rule to remember: if you multiply or divide both sides by a negative number, you need to flip the inequality sign. Keep this in mind, and you'll avoid common mistakes.

Solving 6x < 7

Okay, let's tackle our first inequality: 6x < 7. This one is pretty straightforward. Our goal is to isolate x on one side of the inequality. To do that, we need to get rid of the 6 that's multiplying x. How do we do that? We divide both sides by 6. So, we get:

6x / 6 < 7 / 6

This simplifies to:

x < 7/6

So, the solution to the inequality 6x < 7 is x < 7/6. This means any value of x that is less than 7/6 will satisfy the inequality. If you want to express this as a mixed number, 7/6 is equal to 1 1/6. So, x must be less than 1 1/6.

To visualize this, imagine a number line. You'd put an open circle at 7/6 (or 1 1/6) and shade everything to the left of it. The open circle indicates that 7/6 itself is not included in the solution set because the inequality is strictly less than (x < 7/6), not less than or equal to.

In summary, when solving this inequality, the key step was to divide both sides by a positive number (6), which doesn't require flipping the inequality sign. The solution x < 7/6 represents all the values that make the original inequality true. You can check your work by plugging in a value less than 7/6 into the original inequality to see if it holds true.

Solving 0.7y > -2

Next up, we have 0.7y > -2. Again, our mission is to isolate y. This time, y is being multiplied by 0.7. To get y by itself, we need to divide both sides of the inequality by 0.7. Here’s how it looks:

0. 7y / 0.7 > -2 / 0.7

This simplifies to:

y > -2.8571...

So, the solution to the inequality 0.7y > -2 is approximately y > -2.86 (rounded to two decimal places). This means any value of y greater than -2.86 will satisfy the inequality. Note that since we divided by a positive number (0.7), we didn't need to flip the inequality sign.

To understand this solution better, think about the number line. You would place an open circle at approximately -2.86 and shade everything to the right of it. The open circle signifies that -2.86 is not included in the solution because the inequality is strictly greater than (y > -2.86), not greater than or equal to.

When dealing with decimals, it's often helpful to convert them to fractions to make the division easier. In this case, 0.7 is equal to 7/10. So, the original inequality could be written as (7/10)y > -2. Then, to isolate y, you would multiply both sides by the reciprocal of 7/10, which is 10/7. This would give you the same solution: y > -20/7, which is approximately -2.86.

Remember, the most important thing is to isolate the variable you are solving for by performing inverse operations on both sides of the inequality. And always keep an eye out for negative numbers, because dividing or multiplying by a negative number requires flipping the inequality sign!

Solving -2x < -11

Now, let's solve -2x < -11. This one has a twist! Notice that x is being multiplied by a negative number, -2. This means that when we divide both sides by -2, we have to remember to flip the inequality sign. Let's do it:

-2x / -2 > -11 / -2

Notice how the less than sign (<) has become a greater than sign (>). This simplifies to:

x > 11/2

So, the solution to the inequality -2x < -11 is x > 11/2. This means any value of x that is greater than 11/2 will satisfy the inequality. In decimal form, 11/2 is equal to 5.5. So, x must be greater than 5.5.

To visualize this, imagine a number line. You'd put an open circle at 5.5 and shade everything to the right of it. The open circle indicates that 5.5 itself is not included in the solution set because the inequality is strictly greater than (x > 5.5), not greater than or equal to.

The critical step in this problem was dividing by a negative number. Forgetting to flip the inequality sign is a common mistake, so always double-check when you see a negative coefficient on your variable. The solution x > 11/2 includes all the values that make the original inequality true. As before, you can verify your answer by plugging in a value greater than 5.5 into the original inequality to see if it holds.

Solving -14x > 3.5

Alright, last one! Let's solve -14x > 3.5. Just like the previous problem, we're dealing with a negative coefficient on x. This means we'll need to flip the inequality sign when we divide both sides by -14. Let's do it:

-14x / -14 < 3.5 / -14

See how the greater than sign (>) has become a less than sign (<)? This simplifies to:

x < -0.25

So, the solution to the inequality -14x > 3.5 is x < -0.25. This means any value of x that is less than -0.25 will satisfy the inequality.

On a number line, you would place an open circle at -0.25 and shade everything to the left of it. The open circle indicates that -0.25 is not included in the solution set because the inequality is strictly less than (x < -0.25), not less than or equal to.

The key takeaway here is that dividing by a negative number changes the direction of the inequality. The solution x < -0.25 encompasses all values that make the original inequality true. Always remember to verify your solutions by substituting a value less than -0.25 into the original inequality to ensure it holds valid.

Key Takeaways

  • Isolate the variable: Your primary goal is always to get the variable by itself on one side of the inequality.
  • Flip the sign: Remember to flip the inequality sign whenever you multiply or divide both sides by a negative number.
  • Check your work: Plug your solution back into the original inequality to make sure it holds true.

inequalities can seem tricky at first, but with practice, you'll get the hang of it. Keep these tips in mind, and you'll be solving inequalities like a pro in no time! Good luck, and keep practicing!