Solve The Following Inequality: $-1 \ \textgreater \ -2(x - 4) - 5(4x - 7$\].A. $x \ \textgreater \ 2$ B. $x \ \textgreater \ -23$ C. $x \ \textgreater \ 8$ D. $x \ \textless \ 2$

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Introduction

In this article, we will delve into the world of linear inequalities and explore how to solve them. Linear inequalities are mathematical expressions that contain a variable and a constant, and are often used to model real-world problems. Solving linear inequalities requires a clear understanding of the properties of inequalities and the ability to manipulate them using algebraic techniques.

What are Linear Inequalities?

A linear inequality is a mathematical expression that contains a variable and a constant, and is often written in the form of:

ax + b > c

or

ax + b < c

where a, b, and c are constants, and x is the variable.

Properties of Linear Inequalities

Linear inequalities have several properties that are essential to understand when solving them. These properties include:

  • Addition Property: If a > b, then a + c > b + c.
  • Subtraction Property: If a > b, then a - c > b - c.
  • Multiplication Property: If a > b and c > 0, then ac > bc.
  • Division Property: If a > b and c > 0, then a/c > b/c.

Solving Linear Inequalities

To solve a linear inequality, we need to isolate the variable on one side of the inequality sign. This can be done using the properties of linear inequalities mentioned above.

Example 1: Solving a Simple Linear Inequality

Let's consider the following linear inequality:

2x + 3 > 5

To solve this inequality, we need to isolate the variable x. We can do this by subtracting 3 from both sides of the inequality:

2x > 5 - 3 2x > 2

Next, we can divide both sides of the inequality by 2 to get:

x > 1

Therefore, the solution to the inequality 2x + 3 > 5 is x > 1.

Example 2: Solving a Linear Inequality with a Constant Term

Let's consider the following linear inequality:

3x - 2 > 7

To solve this inequality, we need to isolate the variable x. We can do this by adding 2 to both sides of the inequality:

3x > 7 + 2 3x > 9

Next, we can divide both sides of the inequality by 3 to get:

x > 3

Therefore, the solution to the inequality 3x - 2 > 7 is x > 3.

Example 3: Solving a Linear Inequality with a Negative Coefficient

Let's consider the following linear inequality:

-2x + 5 > 3

To solve this inequality, we need to isolate the variable x. We can do this by subtracting 5 from both sides of the inequality:

-2x > 3 - 5 -2x > -2

Next, we can divide both sides of the inequality by -2 to get:

x < 1

Therefore, the solution to the inequality -2x + 5 > 3 is x < 1.

Example 4: Solving a Linear Inequality with a Negative Constant Term

Let's consider the following linear inequality:

-3x - 2 < 5

To solve this inequality, we need to isolate the variable x. We can do this by adding 2 to both sides of the inequality:

-3x < 5 + 2 -3x < 7

Next, we can divide both sides of the inequality by -3 to get:

x > -7/3

Therefore, the solution to the inequality -3x - 2 < 5 is x > -7/3.

Conclusion

Solving linear inequalities requires a clear understanding of the properties of inequalities and the ability to manipulate them using algebraic techniques. By following the steps outlined in this article, you can solve linear inequalities with ease. Remember to always isolate the variable on one side of the inequality sign and to use the properties of linear inequalities to simplify the inequality.

Practice Problems

  1. Solve the inequality 4x + 2 > 10.
  2. Solve the inequality -2x + 3 < 5.
  3. Solve the inequality x - 2 > 3.
  4. Solve the inequality 2x + 1 < 7.

Answer Key

  1. x > 2
  2. x < 4
  3. x > 5
  4. x < 3

References

  • [1] "Linear Inequalities" by Math Open Reference
  • [2] "Solving Linear Inequalities" by Khan Academy
  • [3] "Linear Inequalities" by Wolfram MathWorld
    Frequently Asked Questions: Solving Linear Inequalities =====================================================

Q: What is a linear inequality?

A: A linear inequality is a mathematical expression that contains a variable and a constant, and is often written in the form of:

ax + b > c

or

ax + b < c

where a, b, and c are constants, and x 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. This can be done using the properties of linear inequalities, such as addition, subtraction, multiplication, and division.

Q: What are the properties of linear inequalities?

A: The properties of linear inequalities include:

  • Addition Property: If a > b, then a + c > b + c.
  • Subtraction Property: If a > b, then a - c > b - c.
  • Multiplication Property: If a > b and c > 0, then ac > bc.
  • Division Property: If a > b and c > 0, then a/c > b/c.

Q: How do I isolate the variable in a linear inequality?

A: To isolate the variable in a linear inequality, you need to perform the following steps:

  1. Add or subtract the same value to both sides of the inequality to get rid of any constants on the same side as the variable.
  2. Multiply or divide both sides of the inequality by the same value to get rid of any coefficients on the same side as the variable.

Q: What is the difference between a linear inequality and a linear equation?

A: A linear equation is a mathematical expression that contains a variable and a constant, and is often 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 a mathematical expression that contains a variable and a constant, and is often written in the form of:

ax + b > c

or

ax + b < c

where a, b, and c are constants, and x is the variable.

Q: Can I use the same methods to solve linear inequalities and linear equations?

A: No, you cannot use the same methods to solve linear inequalities and linear equations. Linear inequalities require a different set of techniques and properties to solve, whereas linear equations can be solved using algebraic methods.

Q: What are some common mistakes to avoid when solving linear inequalities?

A: Some common mistakes to avoid when solving linear inequalities include:

  • Not isolating the variable on one side of the inequality sign.
  • Not using the correct properties of linear inequalities.
  • Not checking the direction of the inequality sign.
  • Not considering the possibility of multiple solutions.

Q: How do I check my work when solving linear inequalities?

A: To check your work when solving linear inequalities, you can:

  • Plug in a test value for the variable and check if the inequality is true.
  • Graph the inequality on a number line and check if the solution is correct.
  • Use a calculator to check if the solution is correct.

Q: What are some real-world applications of linear inequalities?

A: Linear inequalities have many real-world applications, including:

  • Modeling population growth and decline.
  • Determining the maximum or minimum value of a function.
  • Finding the optimal solution to a problem.
  • Analyzing data and making predictions.

Conclusion

Solving linear inequalities requires a clear understanding of the properties of inequalities and the ability to manipulate them using algebraic techniques. By following the steps outlined in this article, you can solve linear inequalities with ease. Remember to always isolate the variable on one side of the inequality sign and to use the properties of linear inequalities to simplify the inequality.

Practice Problems

  1. Solve the inequality 2x + 3 > 5.
  2. Solve the inequality -2x + 3 < 5.
  3. Solve the inequality x - 2 > 3.
  4. Solve the inequality 2x + 1 < 7.

Answer Key

  1. x > 1
  2. x < 4
  3. x > 5
  4. x < 3

References

  • [1] "Linear Inequalities" by Math Open Reference
  • [2] "Solving Linear Inequalities" by Khan Academy
  • [3] "Linear Inequalities" by Wolfram MathWorld