Solve The Inequality:${-3a + 5 \leqslant -16}$

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Introduction

In mathematics, inequalities are a fundamental concept that plays a crucial role in solving various problems. An inequality is a statement that compares two expressions, indicating that one is greater than, less than, or equal to the other. In this article, we will focus on solving linear inequalities, which are inequalities that involve a linear expression on one side and a constant on the other. We will use the given inequality ${-3a + 5 \leqslant -16}$ as an example to demonstrate the steps involved in solving linear inequalities.

What are Linear Inequalities?

Linear inequalities are inequalities that involve a linear expression on one side and a constant on the other. The general form of a linear inequality is ax+b⩽cax + b \leqslant c, where aa, bb, and cc are constants, and xx is the variable. Linear inequalities can be either greater than or equal to (⩾\geqslant) or less than or equal to (⩽\leqslant).

Steps to Solve Linear Inequalities

To solve a linear inequality, we need to isolate the variable on one side of the inequality sign. The steps involved in solving linear inequalities are as follows:

Step 1: Add or Subtract the Same Value to Both Sides

The first step in solving a linear inequality is to add or subtract the same value to both sides of the inequality. This is done to isolate the variable on one side of the inequality sign. For example, in the given inequality ${-3a + 5 \leqslant -16}$, we can add 3a3a to both sides to get 5⩽−16+3a5 \leqslant -16 + 3a.

Step 2: Add or Subtract the Same Value to Both Sides (Continued)

We can continue to add or subtract the same value to both sides of the inequality to isolate the variable. For example, in the previous step, we added 3a3a to both sides. Now, we can add 1616 to both sides to get 21⩽3a21 \leqslant 3a.

Step 3: Divide Both Sides by the Coefficient of the Variable

Once we have isolated the variable on one side of the inequality sign, we can divide both sides by the coefficient of the variable to solve for the variable. For example, in the previous step, we had 21⩽3a21 \leqslant 3a. We can divide both sides by 33 to get 7⩽a7 \leqslant a.

Solving the Given Inequality

Now that we have gone through the steps involved in solving linear inequalities, let's apply these steps to the given inequality ${-3a + 5 \leqslant -16}$.

Step 1: Add or Subtract the Same Value to Both Sides

We can add 3a3a to both sides of the inequality to get 5⩽−16+3a5 \leqslant -16 + 3a.

Step 2: Add or Subtract the Same Value to Both Sides (Continued)

We can add 1616 to both sides of the inequality to get 21⩽3a21 \leqslant 3a.

Step 3: Divide Both Sides by the Coefficient of the Variable

We can divide both sides of the inequality by 33 to get 7⩽a7 \leqslant a.

Conclusion

In this article, we have discussed the concept of linear inequalities and the steps involved in solving them. We have used the given inequality ${-3a + 5 \leqslant -16}$ as an example to demonstrate the steps involved in solving linear inequalities. By following these steps, we can solve linear inequalities and find the solution set.

Frequently Asked Questions

Q: What is a linear inequality?

A: A linear inequality is an inequality that involves a linear expression on one side and a constant on the other.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable on one side of the inequality sign by adding or subtracting the same value to both sides, and then dividing both sides by the coefficient of the variable.

Q: What is the solution set of a linear inequality?

A: The solution set of a linear inequality is the set of all values of the variable that satisfy the inequality.

Example Problems

Problem 1: Solve the inequality 2x+3⩽52x + 3 \leqslant 5

Step 1: Add or Subtract the Same Value to Both Sides

We can subtract 33 from both sides of the inequality to get 2x⩽22x \leqslant 2.

Step 2: Divide Both Sides by the Coefficient of the Variable

We can divide both sides of the inequality by 22 to get x⩽1x \leqslant 1.

Problem 2: Solve the inequality x−2⩾3x - 2 \geqslant 3

Step 1: Add or Subtract the Same Value to Both Sides

We can add 22 to both sides of the inequality to get x⩾5x \geqslant 5.

Step 2: Divide Both Sides by the Coefficient of the Variable

We do not need to divide both sides of the inequality by any value, as the variable is already isolated.

Practice Problems

Problem 1: Solve the inequality 4x−2⩽64x - 2 \leqslant 6

Problem 2: Solve the inequality x+1⩾2x + 1 \geqslant 2

Problem 3: Solve the inequality 2x+1⩽32x + 1 \leqslant 3

Problem 4: Solve the inequality x−1⩾4x - 1 \geqslant 4

Problem 5: Solve the inequality 3x+2⩽53x + 2 \leqslant 5

Conclusion

In this article, we have discussed the concept of linear inequalities and the steps involved in solving them. We have used the given inequality ${-3a + 5 \leqslant -16}$ as an example to demonstrate the steps involved in solving linear inequalities. By following these steps, we can solve linear inequalities and find the solution set. We have also provided example problems and practice problems for readers to practice and reinforce their understanding of linear inequalities.

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Introduction

In our previous article, we discussed the concept of linear inequalities and the steps involved in solving them. However, we understand that readers may still have questions and doubts about linear inequalities. In this article, we will address some of the most frequently asked questions about linear inequalities and provide answers to help clarify any confusion.

Q&A

Q: What is a linear inequality?

A: A linear inequality is an inequality that involves a linear expression on one side and a constant on the other. The general form of a linear inequality is ax+b⩽cax + b \leqslant c, where aa, bb, and cc are constants, and xx is the variable.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable on one side of the inequality sign by adding or subtracting the same value to both sides, and then dividing both sides by the coefficient of the variable.

Q: What is the solution set of a linear inequality?

A: The solution set of a linear inequality is the set of all values of the variable that satisfy the inequality.

Q: Can I use the same steps to solve a linear inequality as I would to solve a linear equation?

A: No, the steps to solve a linear inequality are different from those used to solve a linear equation. When solving a linear inequality, you need to isolate the variable on one side of the inequality sign, but you do not need to find a specific value for the variable.

Q: How do I know if a linear inequality is true or false?

A: To determine if a linear inequality is true or false, you need to test a value of the variable that satisfies the inequality. If the value satisfies the inequality, then the inequality is true. If the value does not satisfy the inequality, then the inequality is false.

Q: Can I use a calculator to solve a linear inequality?

A: Yes, you can use a calculator to solve a linear inequality. However, you need to be careful when using a calculator to solve a linear inequality, as the calculator may not be able to handle the inequality correctly.

Q: How do I graph a linear inequality on a number line?

A: To graph a linear inequality on a number line, you need to plot a point on the number line that satisfies the inequality, and then draw a line on the number line that represents the inequality.

Q: Can I use a graphing calculator to graph a linear inequality?

A: Yes, you can use a graphing calculator to graph a linear inequality. However, you need to be careful when using a graphing calculator to graph a linear inequality, as the calculator may not be able to handle the inequality correctly.

Example Problems

Problem 1: Solve the inequality 2x+3⩽52x + 3 \leqslant 5

Step 1: Add or Subtract the Same Value to Both Sides

We can subtract 33 from both sides of the inequality to get 2x⩽22x \leqslant 2.

Step 2: Divide Both Sides by the Coefficient of the Variable

We can divide both sides of the inequality by 22 to get x⩽1x \leqslant 1.

Problem 2: Solve the inequality x−2⩾3x - 2 \geqslant 3

Step 1: Add or Subtract the Same Value to Both Sides

We can add 22 to both sides of the inequality to get x⩾5x \geqslant 5.

Step 2: Divide Both Sides by the Coefficient of the Variable

We do not need to divide both sides of the inequality by any value, as the variable is already isolated.

Practice Problems

Problem 1: Solve the inequality 4x−2⩽64x - 2 \leqslant 6

Problem 2: Solve the inequality x+1⩾2x + 1 \geqslant 2

Problem 3: Solve the inequality 2x+1⩽32x + 1 \leqslant 3

Problem 4: Solve the inequality x−1⩾4x - 1 \geqslant 4

Problem 5: Solve the inequality 3x+2⩽53x + 2 \leqslant 5

Conclusion

In this article, we have addressed some of the most frequently asked questions about linear inequalities and provided answers to help clarify any confusion. We have also provided example problems and practice problems for readers to practice and reinforce their understanding of linear inequalities. By following the steps outlined in this article, readers should be able to solve linear inequalities and understand the concept of linear inequalities.

Frequently Asked Questions

Q: What is a linear inequality?

A: A linear inequality is an inequality that involves a linear expression on one side and a constant on the other.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable on one side of the inequality sign by adding or subtracting the same value to both sides, and then dividing both sides by the coefficient of the variable.

Q: What is the solution set of a linear inequality?

A: The solution set of a linear inequality is the set of all values of the variable that satisfy the inequality.

Q: Can I use the same steps to solve a linear inequality as I would to solve a linear equation?

A: No, the steps to solve a linear inequality are different from those used to solve a linear equation. When solving a linear inequality, you need to isolate the variable on one side of the inequality sign, but you do not need to find a specific value for the variable.

Q: How do I know if a linear inequality is true or false?

A: To determine if a linear inequality is true or false, you need to test a value of the variable that satisfies the inequality. If the value satisfies the inequality, then the inequality is true. If the value does not satisfy the inequality, then the inequality is false.

Q: Can I use a calculator to solve a linear inequality?

A: Yes, you can use a calculator to solve a linear inequality. However, you need to be careful when using a calculator to solve a linear inequality, as the calculator may not be able to handle the inequality correctly.

Q: How do I graph a linear inequality on a number line?

A: To graph a linear inequality on a number line, you need to plot a point on the number line that satisfies the inequality, and then draw a line on the number line that represents the inequality.

Q: Can I use a graphing calculator to graph a linear inequality?

A: Yes, you can use a graphing calculator to graph a linear inequality. However, you need to be careful when using a graphing calculator to graph a linear inequality, as the calculator may not be able to handle the inequality correctly.