Solving Inequalities: Find The Solution

Oct 17, 2025 by ADMIN 40 views

Alright, let's break down how to solve this inequality and figure out which of the given numbers are actually solutions. Inequalities might seem a bit tricky at first, but once you get the hang of the rules, they're pretty straightforward. We will solve the inequality step by step and then check each of the provided numbers to see if they fit the solution. So, let's dive right in!

Solving the Inequality: $-54 \leq -6c$

When solving inequalities, our main goal is the same as when solving equations: to isolate the variable. In this case, we want to get c by itself on one side of the inequality. Here's the inequality we're starting with:

$-54 \leq -6c$

To isolate c, we need to get rid of the -6 that's multiplying it. We can do this by dividing both sides of the inequality by -6. Now, here's a super important rule to remember:

When you multiply or divide both sides of an inequality by a negative number, you have to flip the direction of the inequality sign.

So, when we divide both sides by -6, we get:

$\frac{-54}{-6} \geq \frac{-6c}{-6}$

Notice that the $\leq$ sign has changed to $\geq$ because we divided by a negative number. Now, let's simplify:

$9 \geq c$

This inequality can also be written as:

$c \leq 9$

This means that c is less than or equal to 9. In other words, any number that is 9 or smaller is a solution to the inequality. Now that we've solved the inequality, let's check which of the given numbers satisfy this condition.

Checking the Numbers

We have a list of numbers to check: 5, 12, 0, 15, and 9. We need to see which of these numbers are less than or equal to 9.

1. Checking 5

Is 5 less than or equal to 9? Yes, it is! So, 5 is a solution to the inequality.

$5 \leq 9$ (True)

2. Checking 12

Is 12 less than or equal to 9? No, it is not! 12 is greater than 9. So, 12 is not a solution.

$12 \leq 9$ (False)

3. Checking 0

Is 0 less than or equal to 9? Yes, definitely! 0 is much smaller than 9, so it's a solution.

$0 \leq 9$ (True)

4. Checking 15

Is 15 less than or equal to 9? Nope, 15 is way bigger than 9. So, 15 is not a solution.

$15 \leq 9$ (False)

5. Checking 9

Is 9 less than or equal to 9? Yes, it is! 9 is equal to 9, and the inequality includes the "equal to" part. So, 9 is a solution.

$9 \leq 9$ (True)

Conclusion

After solving the inequality $-54 \leq -6c$ and checking the given numbers, we found that the solutions are 5, 0, and 9. Remember, the key to solving inequalities is to isolate the variable while paying close attention to the direction of the inequality sign, especially when multiplying or dividing by a negative number.

So, to recap:

The solution to the inequality $c \leq 9$
The numbers that are solutions from the list are: 5, 0, and 9.

And there you have it! Solving inequalities can be super useful in all sorts of real-world problems, from figuring out budget constraints to understanding physical limitations. Keep practicing, and you'll become a pro in no time!

Further Practice

To solidify your understanding, try solving these additional inequality problems:

Solve: $-3x > 12$ . Which of the following are solutions: -5, -4, -3, 0?
Solve: $2y + 5 \leq 11$ . Which of the following are solutions: 0, 3, 6, 10?
Solve: $-4z - 8 \geq 4$ . Which of the following are solutions: -5, -3, 0, 2?

Remember to flip the inequality sign when multiplying or dividing by a negative number. Good luck, and happy solving!

Tips and Tricks for Solving Inequalities

Alright guys, let's boost your inequality-solving skills with some handy tips and tricks. These strategies can make things easier and help you avoid common mistakes.

1. Treat Inequalities Like Equations (Mostly)

For the most part, you can treat inequalities just like equations. You can add, subtract, multiply, and divide both sides to isolate the variable. This makes the initial steps of solving inequalities feel familiar and straightforward.

Example: Consider the inequality $x + 5 < 10$ . To solve for x, you can subtract 5 from both sides, just like you would do with an equation:

$x + 5 - 5 < 10 - 5$

Which simplifies to:

$x < 5$

2. The Flip Rule: Remember It!

As we highlighted earlier, the most critical rule to remember is that when you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality sign. This is essential for getting the correct solution. Forgetting this rule is a common mistake, so always double-check when you're dealing with negative numbers.

Example: Solve $-2y \geq 8$ . To isolate y, divide both sides by -2. Remember to flip the inequality sign:

$\frac{-2y}{-2} \leq \frac{8}{-2}$

Which simplifies to:

$y \leq -4$

3. Simplify First

Before you start isolating the variable, simplify both sides of the inequality as much as possible. This might involve combining like terms, distributing, or clearing fractions. Simplifying first makes the problem easier to manage and reduces the chance of errors.

Example: Solve $3(x + 2) - 5 > 4$ . First, distribute the 3:

$3x + 6 - 5 > 4$

Then, combine like terms:

$3x + 1 > 4$

Now, it’s much easier to isolate x.

4. Visualize on a Number Line

Sometimes, it helps to visualize the solution on a number line. This is especially useful for understanding what the inequality actually means. For example, if you have $x > 3$ , you can draw a number line with an open circle at 3 and shade everything to the right.

Open circle: The number is not included in the solution.
Closed circle: The number is included in the solution.
Shading: Represents all the numbers that satisfy the inequality.

5. Check Your Solution

After you've solved the inequality, check your solution by plugging a number from your solution set back into the original inequality. This helps you confirm that your solution is correct.

Example: Suppose you solved $2x < 6$ and found $x < 3$ . Pick a number less than 3, like 0, and plug it into the original inequality:

$2(0) < 6$

$0 < 6$ (True)

Since the inequality holds true, your solution is likely correct.

6. Watch Out for Special Cases

Be aware of special cases where the inequality results in a statement that is always true or always false. These cases indicate that either all real numbers are solutions or there are no solutions.

Example 1: Solve $x + 1 < x + 2$ . Subtract x from both sides:

$1 < 2$

This statement is always true, so all real numbers are solutions.

Example 2: Solve $x - 1 > x$ . Subtract x from both sides:

$-1 > 0$

This statement is always false, so there are no solutions.

7. Practice Regularly

The best way to improve your inequality-solving skills is to practice regularly. The more problems you solve, the more comfortable you'll become with the different types of inequalities and the techniques for solving them.

By keeping these tips and tricks in mind, you'll be well-equipped to tackle any inequality problem that comes your way. Happy solving!