What Value Of $x$ Is In The Solution Set Of $3(x-4) \geq 5x+2$?A. -10 B. -5 C. 5 D. 10
Introduction
In mathematics, solving inequalities is a crucial concept that helps us understand the relationship between different variables. In this article, we will focus on solving the inequality $3(x-4) \geq 5x+2$ and find the value of $x$ that is in the solution set.
Understanding the Inequality
The given inequality is $3(x-4) \geq 5x+2$. To solve this inequality, we need to isolate the variable $x$. We can start by expanding the left-hand side of the inequality using the distributive property.
Expanding the Left-Hand Side
Using the distributive property, we can expand the left-hand side of the inequality as follows:
So, the inequality becomes:
Isolating the Variable $x$
Now, we need to isolate the variable $x$ by moving all the terms involving $x$ to one side of the inequality. We can do this by subtracting $3x$ from both sides of the inequality.
Simplifying the Inequality
Next, we need to simplify the inequality by combining like terms. We can do this by subtracting $2$ from both sides of the inequality.
Solving for $x$
Now, we need to solve for $x$ by dividing both sides of the inequality by $2$. However, we need to be careful when dividing both sides of an inequality by a negative number. In this case, we are dividing by $2$, which is a positive number.
Conclusion
In conclusion, the value of $x$ that is in the solution set of the inequality $3(x-4) \geq 5x+2$ is $x \leq -7$. This means that any value of $x$ that is less than or equal to $-7$ is in the solution set.
Solution Set
The solution set of the inequality $3(x-4) \geq 5x+2$ is the set of all values of $x$ that satisfy the inequality. In this case, the solution set is $x \leq -7$.
Graphical Representation
The solution set of the inequality $3(x-4) \geq 5x+2$ can be represented graphically on a number line. The number line is divided into two parts: one part represents the values of $x$ that are less than or equal to $-7$, and the other part represents the values of $x$ that are greater than $-7$.
Final Answer
The final answer is $x \leq -7$. This means that any value of $x$ that is less than or equal to $-7$ is in the solution set.
Comparison with Options
Now, let's compare the solution set $x \leq -7$ with the given options:
A. -10 B. -5 C. 5 D. 10
We can see that the value $x = -10$ is in the solution set, as it is less than or equal to $-7$. Therefore, the correct answer is:
A. -10
Final Conclusion
In conclusion, the value of $x$ that is in the solution set of the inequality $3(x-4) \geq 5x+2$ is $x \leq -7$. This means that any value of $x$ that is less than or equal to $-7$ is in the solution set. The correct answer is:
A. -10
Introduction
In the previous article, we discussed how to solve the inequality $3(x-4) \geq 5x+2$ and found the value of $x$ that is in the solution set. In this article, we will answer some frequently asked questions (FAQs) about solving inequalities.
Q: What is the difference between solving an equation and solving an inequality?
A: Solving an equation involves finding the value of a variable that makes the equation true, whereas solving an inequality involves finding the set of values of a variable that make the inequality true.
Q: How do I know which direction to move the terms when solving an inequality?
A: When solving an inequality, you need to move the terms in the same direction as the inequality sign. For example, if the inequality is $x + 2 > 5$, you would subtract 2 from both sides to get $x > 3$.
Q: What is the difference between a strict inequality and a non-strict inequality?
A: A strict inequality is an inequality that is written with a strict inequality sign, such as $x > 3$ or $x < 5$. A non-strict inequality is an inequality that is written with a non-strict inequality sign, such as $x \geq 3$ or $x \leq 5$.
Q: How do I graph an inequality on a number line?
A: To graph an inequality on a number line, you need to draw a line at the value that is being compared to the variable. If the inequality is strict, you need to draw an open circle at the value. If the inequality is non-strict, you need to draw a closed circle at the value.
Q: Can I use the same steps to solve a system of inequalities as I would to solve a system of equations?
A: No, you cannot use the same steps to solve a system of inequalities as you would to solve a system of equations. When solving a system of inequalities, you need to find the intersection of the solution sets of each inequality.
Q: How do I find the intersection of two solution sets?
A: To find the intersection of two solution sets, you need to find the values that are common to both solution sets. You can do this by graphing the two solution sets on a number line and finding the values that are shaded in both graphs.
Q: Can I use a calculator to solve an inequality?
A: Yes, you can use a calculator to solve an inequality. However, you need to be careful when using a calculator to solve an inequality, as it may not always give you the correct solution.
Q: How do I check my solution to an inequality?
A: To check your solution to an inequality, you need to plug the value back into the original inequality and see if it is true. If it is true, then your solution is correct. If it is not true, then your solution is incorrect.
Q: Can I use the same steps to solve a quadratic inequality as I would to solve a quadratic equation?
A: No, you cannot use the same steps to solve a quadratic inequality as you would to solve a quadratic equation. When solving a quadratic inequality, you need to use a different method, such as factoring or using the quadratic formula.
Q: How do I factor a quadratic expression?
A: To factor a quadratic expression, you need to find two binomials whose product is equal to the quadratic expression. You can do this by looking for two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term.
Q: Can I use a graphing calculator to solve a quadratic inequality?
A: Yes, you can use a graphing calculator to solve a quadratic inequality. However, you need to be careful when using a graphing calculator to solve a quadratic inequality, as it may not always give you the correct solution.
Q: How do I use a graphing calculator to solve a quadratic inequality?
A: To use a graphing calculator to solve a quadratic inequality, you need to enter the quadratic expression into the calculator and use the "solve" function to find the solution set.
Q: Can I use a graphing calculator to solve a system of inequalities?
A: Yes, you can use a graphing calculator to solve a system of inequalities. However, you need to be careful when using a graphing calculator to solve a system of inequalities, as it may not always give you the correct solution.
Q: How do I use a graphing calculator to solve a system of inequalities?
A: To use a graphing calculator to solve a system of inequalities, you need to enter each inequality into the calculator and use the "solve" function to find the solution set. You can then find the intersection of the solution sets to find the final solution.
Conclusion
In conclusion, solving inequalities is an important concept in mathematics that involves finding the set of values of a variable that make the inequality true. By following the steps outlined in this article, you can solve inequalities and find the solution set. Additionally, you can use a graphing calculator to solve inequalities and find the solution set.