1. Solve For \[$ X \$\]: \[$\frac{x}{6} + 11 \ \textgreater \ 15\$\]2. Solve For \[$ X \$\]: \[$-14x + 1 \geq 71\$\]3. Solve For \[$ X \$\]: \[$-3x - 1 \geq 41\$\]
In mathematics, linear inequalities are a fundamental concept that plays a crucial role in solving various problems in algebra, geometry, and other branches of mathematics. A linear inequality is an inequality that can be written in the form of ax + b > c, where a, b, and c are constants, and x is the variable. In this article, we will focus on solving three linear inequalities of the form ax + b > c, where a, b, and c are constants, and x is the variable.
Solving the First Inequality: x/6 + 11 > 15
To solve the first inequality, we need to isolate the variable x. The given inequality is x/6 + 11 > 15. Our goal is to get x by itself on one side of the inequality.
Step 1: Subtract 11 from both sides
Subtracting 11 from both sides of the inequality, we get:
x/6 > 15 - 11
x/6 > 4
Step 2: Multiply both sides by 6
Multiplying both sides of the inequality by 6, we get:
x > 4 * 6
x > 24
Therefore, the solution to the first inequality is x > 24.
Solving the Second Inequality: -14x + 1 ≥ 71
To solve the second inequality, we need to isolate the variable x. The given inequality is -14x + 1 ≥ 71. Our goal is to get x by itself on one side of the inequality.
Step 1: Subtract 1 from both sides
Subtracting 1 from both sides of the inequality, we get:
-14x ≥ 71 - 1
-14x ≥ 70
Step 2: Divide both sides by -14
Dividing both sides of the inequality by -14, we get:
x ≤ 70 / -14
x ≤ -5
Therefore, the solution to the second inequality is x ≤ -5.
Solving the Third Inequality: -3x - 1 ≥ 41
To solve the third inequality, we need to isolate the variable x. The given inequality is -3x - 1 ≥ 41. Our goal is to get x by itself on one side of the inequality.
Step 1: Add 1 to both sides
Adding 1 to both sides of the inequality, we get:
-3x ≥ 41 + 1
-3x ≥ 42
Step 2: Divide both sides by -3
Dividing both sides of the inequality by -3, we get:
x ≤ 42 / -3
x ≤ -14
Therefore, the solution to the third inequality is x ≤ -14.
Conclusion
In this article, we have solved three linear inequalities of the form ax + b > c, where a, b, and c are constants, and x is the variable. We have used the steps of subtracting, adding, multiplying, and dividing to isolate the variable x and find the solution to each inequality. The solutions to the three inequalities are x > 24, x ≤ -5, and x ≤ -14, respectively.
Tips and Tricks
When solving linear inequalities, it is essential to follow the order of operations (PEMDAS) and to be careful when multiplying or dividing both sides of the inequality by a negative number. Additionally, it is crucial to check the solution by plugging it back into the original inequality to ensure that it is true.
Practice Problems
To practice solving linear inequalities, try the following problems:
- Solve for x: 2x + 5 > 11
- Solve for x: -2x - 3 ≥ 13
- Solve for x: x/4 + 2 > 7
In this article, we will answer some frequently asked questions about linear inequalities. Whether you are a student, a teacher, or simply someone who wants to learn more about linear inequalities, this article is for you.
Q: What is a linear inequality?
A linear inequality is an inequality that can be written in the form of ax + b > c, where a, b, and c are constants, and x is the variable.
Q: How do I solve a linear inequality?
To solve a linear inequality, you need to isolate the variable x. You can do this by adding, subtracting, multiplying, or dividing both sides of the inequality by a constant. However, when multiplying or dividing both sides of the inequality by a negative number, you need to flip the direction of the inequality.
Q: What is the difference between a linear inequality and a linear equation?
A linear equation is an equation that can be written in the form of ax + b = c, where a, b, and c are constants, and x is the variable. A linear inequality, on the other hand, is an inequality that can be written in the form of ax + b > c, where a, b, and c are constants, and x is the variable.
Q: How do I know which direction to flip the inequality when multiplying or dividing by a negative number?
When multiplying or dividing both sides of the inequality by a negative number, you need to flip the direction of the inequality. For example, if you have the inequality x > 5 and you multiply both sides by -2, you need to flip the direction of the inequality to get -2x < -10.
Q: Can I use the same steps to solve a linear inequality as I would to solve a linear equation?
No, you cannot use the same steps to solve a linear inequality as you would to solve a linear equation. When solving a linear inequality, you need to isolate the variable x and flip the direction of the inequality when multiplying or dividing by a negative number.
Q: What are some common mistakes to avoid when solving linear inequalities?
Some common mistakes to avoid when solving linear inequalities include:
- Not flipping the direction of the inequality when multiplying or dividing by a negative number
- Not isolating the variable x
- Not checking the solution by plugging it back into the original inequality
Q: How do I check my solution to a linear inequality?
To check your solution to a linear inequality, you need to plug it back into the original inequality and make sure that it is true. If the solution is not true, you need to go back and recheck your work.
Q: What are some real-world applications of linear inequalities?
Linear inequalities have many real-world applications, including:
- Budgeting and finance
- Science and engineering
- Economics and business
- Computer science and programming
Conclusion
In this article, we have answered some frequently asked questions about linear inequalities. Whether you are a student, a teacher, or simply someone who wants to learn more about linear inequalities, this article is for you. Remember to always follow the steps outlined in this article and to be careful when solving linear inequalities.
Practice Problems
To practice solving linear inequalities, try the following problems:
- Solve for x: 2x + 5 > 11
- Solve for x: -2x - 3 ≥ 13
- Solve for x: x/4 + 2 > 7
Remember to follow the steps outlined in this article and to be careful when solving the inequalities.