Solving Inequality: -5x - 5 ≤ 15

Hey guys! Let's break down this inequality problem step-by-step. Inequalities might seem a bit tricky at first, but once you understand the basic principles, they become super manageable. In this article, we're going to solve the inequality $-5x - 5 \le 15$ for $x$. This involves isolating $x$ on one side of the inequality to find the range of values that satisfy the condition. So, grab your favorite beverage, and let's dive in!

Understanding Inequalities

Before we jump into solving the specific inequality, let’s briefly talk about what inequalities are and how they differ from equations. An equation uses an equals sign (=) to show that two expressions are equal. An inequality, on the other hand, uses symbols like less than (<), greater than (>), less than or equal to ($\le$), or greater than or equal to ($\ge$) to show the relationship between two expressions. Solving an inequality means finding all the values of the variable that make the inequality true.

When dealing with inequalities, there are a few important rules to keep in mind. The most crucial rule is that when you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign. This is because multiplying or dividing by a negative number changes the sign of the expressions, and to maintain the correct relationship, you need to flip the inequality sign. For example, if $a < b$, then $-a > -b$. This rule is essential for correctly solving inequalities, especially when the variable is multiplied by a negative coefficient.

Additionally, it’s good to remember that you can add or subtract the same number from both sides of an inequality without changing the direction of the inequality sign. This is similar to how you would manipulate equations. For instance, if $a < b$, then $a + c < b + c$ and $a - c < b - c$. These basic operations allow you to isolate the variable and simplify the inequality, bringing you closer to the solution. Keeping these principles in mind will make solving inequalities much easier and more intuitive. Understanding these rules ensures that you can accurately manipulate and solve a wide variety of inequality problems.

Step-by-Step Solution

Let's solve the inequality $-5x - 5 \le 15$ for $x$. Here’s a detailed, step-by-step explanation to guide you through the process. Our main goal is to isolate $x$ on one side of the inequality to determine the range of values that satisfy the condition.

Step 1: Add 5 to Both Sides

To start, we want to isolate the term with $x$. We can do this by adding 5 to both sides of the inequality. This operation maintains the balance of the inequality and helps us move closer to isolating $x$.

$-5x - 5 + 5 \le 15 + 5$

Simplifying this gives us:

$-5x \le 20$

Step 2: Divide Both Sides by -5

Now, we need to get $x$ by itself. Since $x$ is multiplied by -5, we will divide both sides of the inequality by -5. Remember the crucial rule: when you divide (or multiply) an inequality by a negative number, you must reverse the direction of the inequality sign. This is a critical step to ensure we get the correct solution.

$\frac{-5x}{-5} \ge \frac{20}{-5}$

Notice that the $\le$ sign has changed to $\ge$ because we divided by a negative number.

Step 3: Simplify

After dividing, we simplify the inequality:

$x \ge -4$

This means that $x$ is greater than or equal to -4. In other words, any value of $x$ that is -4 or larger will satisfy the original inequality $-5x - 5 \le 15$.

Step 4: Express the Solution

We can express the solution in a few different ways. One way is using inequality notation, which we already have: $x \ge -4$. Another way is using interval notation. In interval notation, the solution is written as $[-4, \infty)$. This notation indicates that the solution includes all numbers from -4 (inclusive, indicated by the square bracket) to positive infinity.

Summary of Steps

1. Add 5 to both sides: $-5x - 5 + 5 \le 15 + 5$ becomes $-5x \le 20$.
2. Divide both sides by -5 (and reverse the inequality sign): $\frac{-5x}{-5} \ge \frac{20}{-5}$.
3. Simplify: $x \ge -4$.

Following these steps carefully ensures that you arrive at the correct solution. Remember, the key is to isolate the variable while paying close attention to the rules for manipulating inequalities.

Verification

To make sure our solution is correct, we can test a value of $x$ that satisfies the inequality $x \ge -4$. Let’s pick $x = 0$ since 0 is greater than -4. Plug $x = 0$ into the original inequality:

$-5(0) - 5 \le 15$

Simplifying, we get:

$-5 \le 15$

This is true, so our solution is likely correct. Now, let’s test a value that does not satisfy the inequality, such as $x = -5$:

$-5(-5) - 5 \le 15$

Simplifying, we get:

$25 - 5 \le 15$

$20 \le 15$

This is false, which further confirms that our solution $x \ge -4$ is correct. Testing values helps ensure that you haven’t made any mistakes in your calculations and that the solution accurately represents the range of values that satisfy the original inequality. Always take the time to verify your solution to increase confidence in your answer!

Common Mistakes to Avoid

When solving inequalities, it’s easy to make a few common mistakes. Recognizing these pitfalls can save you from errors and help you approach inequality problems with greater confidence. Let's go through some of these common mistakes:

Forgetting to Reverse the Inequality Sign

The most frequent mistake is forgetting to reverse the inequality sign when multiplying or dividing both sides by a negative number. As we discussed earlier, this is a crucial step. If you miss this, your solution will be completely wrong. Always double-check whether you’ve multiplied or divided by a negative number and, if so, ensure you flip the inequality sign. This simple check can make a big difference in the accuracy of your answer.

Incorrectly Applying Arithmetic Operations

Another common mistake is making errors in basic arithmetic. This can include adding, subtracting, multiplying, or dividing incorrectly. Even a small arithmetic error can throw off the entire solution. To avoid this, take your time and double-check each step. If necessary, use a calculator to verify your calculations, especially when dealing with more complex numbers. Accuracy in arithmetic is essential for arriving at the correct solution.

Not Distributing Properly

When dealing with inequalities that involve parentheses, it’s important to distribute properly. Make sure you multiply each term inside the parentheses by the term outside. For example, if you have $2(x + 3) < 10$, you need to distribute the 2 to both $x$ and 3, resulting in $2x + 6 < 10$. Failing to distribute correctly can lead to an incorrect inequality and, consequently, an incorrect solution. Always take the time to carefully distribute and double-check your work.

Misinterpreting the Inequality Sign

Sometimes, students misinterpret the inequality signs, especially when they are used in combination with negative numbers. Make sure you understand the difference between “less than” (<), “greater than” (>), “less than or equal to” ($\le$), and “greater than or equal to” ($\ge$). A good way to avoid confusion is to read the inequality from left to right, paying close attention to the direction of the sign. For example, $x \ge -4$ means “$x$ is greater than or equal to -4.”

Not Checking the Solution

Finally, one of the biggest mistakes is not checking the solution. As we demonstrated earlier, plugging a value that satisfies your solution back into the original inequality can help you verify whether your answer is correct. If the inequality holds true, then your solution is likely correct. If it doesn’t, you know you’ve made a mistake somewhere and need to go back and review your steps. Always take the time to check your solution – it’s a simple step that can save you from submitting an incorrect answer.

By being aware of these common mistakes and taking steps to avoid them, you can improve your accuracy and confidence when solving inequalities. Remember to double-check your work, pay attention to the details, and always verify your solution.

Conclusion

Alright, folks! We’ve successfully solved the inequality $-5x - 5 \le 15$. Remember, the key steps are to isolate the variable $x$ by performing the same operations on both sides of the inequality, and to reverse the inequality sign when multiplying or dividing by a negative number. Always verify your solution to ensure accuracy.

Keep practicing, and you’ll become a pro at solving inequalities in no time! You got this! Stay tuned for more math tips and tricks. Peace out!