Solve $54 - 10x \leq 20 + 7x$.A. $x \leq 2$ B. $ X ≥ 2 X \geq 2 X ≥ 2 [/tex] C. $x \geq -2$ D. $x \leq -2$
Introduction
Linear inequalities are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving a specific linear inequality, which is given by the equation $54 - 10x \leq 20 + 7x$. We will break down the solution step by step, using clear and concise language to ensure that readers understand the process.
Understanding Linear Inequalities
Before we dive into the solution, let's take a moment to understand what linear inequalities are. A linear inequality is an inequality that involves a linear expression, which is an expression that can be written in the form $ax + b$, where $a$ and $b$ are constants, and $x$ is the variable. Linear inequalities can be either equalities or inequalities, and they can be written in the form $ax + b \leq c$ or $ax + b \geq c$, where $c$ is a constant.
Solving the Linear Inequality
Now that we have a basic understanding of linear inequalities, let's move on to solving the given inequality. The inequality we are given is $54 - 10x \leq 20 + 7x$. Our goal is to isolate the variable $x$ on one side of the inequality.
Step 1: Add $10x$ to Both Sides
To start solving the inequality, we need to get all the terms involving $x$ on one side of the inequality. We can do this by adding $10x$ to both sides of the inequality. This gives us:
Simplifying the left-hand side, we get:
Step 2: Subtract 20 from Both Sides
Next, we need to get rid of the constant term on the right-hand side of the inequality. We can do this by subtracting 20 from both sides of the inequality. This gives us:
Simplifying the left-hand side, we get:
Step 3: Divide Both Sides by 17
Finally, we need to isolate the variable $x$ on one side of the inequality. We can do this by dividing both sides of the inequality by 17. This gives us:
Simplifying the left-hand side, we get:
Conclusion
In conclusion, we have solved the linear inequality $54 - 10x \leq 20 + 7x$. By following the steps outlined above, we have isolated the variable $x$ on one side of the inequality, and we have found that $x \leq 2$. This is the correct solution to the inequality.
Answer
The correct answer is:
A. $x \leq 2$
Final Thoughts
Introduction
In our previous article, we covered the basics of solving linear inequalities, including the step-by-step process for solving a specific inequality. However, we know that there are many more questions and concerns that students may have when it comes to solving linear inequalities. In this article, we will address some of the most frequently asked questions about solving linear inequalities.
Q: What is the difference between a linear inequality and a linear equation?
A: A linear equation is an equation that involves a linear expression, which is an expression that can be written in the form $ax + b$, where $a$ and $b$ are constants, and $x$ is the variable. A linear inequality, on the other hand, is an inequality that involves a linear expression. The key difference between the two is that a linear equation is an equality, while a linear inequality is an inequality.
Q: How do I know which direction to move the inequality sign when I add or subtract the same value to both sides?
A: When you add or subtract the same value to both sides of an inequality, the direction of the inequality sign remains the same. For example, if you have the inequality $x < 5$ and you add 3 to both sides, the inequality becomes $x + 3 < 8$. The direction of the inequality sign remains the same, which is less than.
Q: Can I multiply or divide both sides of an inequality by a negative number?
A: No, you cannot multiply or divide both sides of an inequality by a negative number. When you multiply or divide both sides of an inequality by a negative number, the direction of the inequality sign changes. For example, if you have the inequality $x < 5$ and you multiply both sides by -1, the inequality becomes $-x > -5$. The direction of the inequality sign has changed, which is greater than.
Q: How do I know which direction to move the inequality sign when I multiply or divide both sides by a fraction?
A: When you multiply or divide both sides of an inequality by a fraction, the direction of the inequality sign remains the same if the fraction is positive, and changes if the fraction is negative. For example, if you have the inequality $x < 5$ and you multiply both sides by $\frac{1}{2}$, the inequality becomes $\frac{1}{2}x < \frac{5}{2}$. The direction of the inequality sign remains the same, which is less than.
Q: Can I use the same steps to solve a linear inequality as I would to solve a linear equation?
A: Yes, you can use the same steps to solve a linear inequality as you would to solve a linear equation. The steps include adding or subtracting the same value to both sides, multiplying or dividing both sides by the same value, and isolating the variable on one side of the inequality.
Q: What if I have a linear inequality with a variable on both sides?
A: If you have a linear inequality with a variable on both sides, you can add or subtract the same value to both sides to get the variable on one side of the inequality. For example, if you have the inequality $x + 2 < 5$, you can subtract 2 from both sides to get $x < 3$.
Conclusion
In conclusion, solving linear inequalities can be a challenging task, but with practice and patience, it can become second nature. We hope that this article has addressed some of the most frequently asked questions about solving linear inequalities and has provided you with a better understanding of the process.
Final Thoughts
Remember, solving linear inequalities is all about following the steps and using the correct operations to isolate the variable on one side of the inequality. With practice and patience, you will become proficient in solving linear inequalities and will be able to tackle even the most challenging problems with confidence.