Write A Situation For The Inequality $15x - 20 \leq 130$ And Solve It.

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Introduction

Linear inequalities are a fundamental concept in mathematics, and they play a crucial role in various fields such as economics, engineering, and computer science. In this article, we will focus on solving linear inequalities, specifically the inequality $15x - 20 \leq 130$. We will start by creating a real-world situation that illustrates this inequality and then proceed to solve it step by step.

Situation

Suppose a company is planning to produce a new product, and they need to determine the maximum number of units they can produce within a certain budget. The company has a fixed cost of $20 per unit, and they want to spend no more than $130 on production costs. If they sell each unit for $15, how many units can they produce within their budget?

Mathematically, this situation can be represented by the inequality $15x - 20 \leq 130$, where x is the number of units produced.

Solving the Inequality

To solve the inequality $15x - 20 \leq 130$, we need to isolate the variable x. We can start by adding 20 to both sides of the inequality:

15x−20+20≤130+2015x - 20 + 20 \leq 130 + 20

This simplifies to:

15x≤15015x \leq 150

Next, we can divide both sides of the inequality by 15:

15x15≤15015\frac{15x}{15} \leq \frac{150}{15}

This simplifies to:

x≤10x \leq 10

Interpretation

The solution to the inequality $15x - 20 \leq 130$ is x ≤ 10. This means that the company can produce no more than 10 units within their budget of $130.

Graphical Representation

We can also represent the solution to the inequality graphically. The inequality $15x - 20 \leq 130$ can be written as $15x \leq 150$, which can be represented by a line on a number line. The line represents the boundary of the solution set, and the region to the left of the line represents the solution set.

Conclusion

In this article, we created a real-world situation that illustrates the inequality $15x - 20 \leq 130$ and then solved it step by step. We isolated the variable x and found that the solution to the inequality is x ≤ 10. This means that the company can produce no more than 10 units within their budget of $130.

Tips and Tricks

  • When solving linear inequalities, it's essential to isolate the variable on one side of the inequality.
  • When adding or subtracting a constant from both sides of an inequality, make sure to keep the direction of the inequality the same.
  • When multiplying or dividing both sides of an inequality by a negative number, make sure to reverse the direction of the inequality.

Practice Problems

  1. Solve the inequality $2x + 5 \leq 17$.
  2. Solve the inequality $x - 3 \geq 2$.
  3. Solve the inequality $4x - 2 \leq 12$.

Solutions

  1. 2x+5≤172x + 5 \leq 17

    2x+5−5≤17−52x + 5 - 5 \leq 17 - 5

    2x≤122x \leq 12

    2x2≤122\frac{2x}{2} \leq \frac{12}{2}

    x≤6x \leq 6

  2. x−3≥2x - 3 \geq 2

    x−3+3≥2+3x - 3 + 3 \geq 2 + 3

    x≥5x \geq 5

  3. 4x−2≤124x - 2 \leq 12

    4x−2+2≤12+24x - 2 + 2 \leq 12 + 2

    4x≤144x \leq 14

    4x4≤144\frac{4x}{4} \leq \frac{14}{4}

    x≤3.5x \leq 3.5

Conclusion

In this article, we solved the inequality $15x - 20 \leq 130$ and created a real-world situation that illustrates this inequality. We also provided tips and tricks for solving linear inequalities and included practice problems with solutions. We hope this article has been helpful in understanding how to solve linear inequalities.

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Introduction

Linear inequalities are a fundamental concept in mathematics, and they play a crucial role in various fields such as economics, engineering, and computer science. In this article, we will answer some of the most frequently asked questions about linear inequalities.

Q&A

Q: What is a linear inequality?

A: A linear inequality is an inequality that can be written in the form 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. You can do this by adding or subtracting a constant from both sides of the inequality, or by multiplying or dividing both sides of the inequality by a non-zero constant.

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

A: A linear equation is an equation that can be written in the form 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 ax + b ≤ c, where a, b, and c are constants, and x is the variable.

Q: How do I graph a linear inequality?

A: To graph a linear inequality, you need to graph the corresponding linear equation and then shade the region on one side of the line. If the inequality is of the form ax + b ≤ c, you shade the region below the line. If the inequality is of the form ax + b ≥ c, you shade the region above the line.

Q: Can I solve a linear inequality by using algebraic methods?

A: Yes, you can solve a linear inequality by using algebraic methods. You can add or subtract a constant from both sides of the inequality, or you can multiply or divide both sides of the inequality by a non-zero constant.

Q: Can I solve a linear inequality by using graphical methods?

A: Yes, you can solve a linear inequality by using graphical methods. You can graph the corresponding linear equation and then shade the region on one side of the line.

Q: What is the solution to a linear inequality?

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

Q: How do I find the solution to a linear inequality?

A: To find the solution to a linear inequality, you need to isolate the variable on one side of the inequality. You can do this by adding or subtracting a constant from both sides of the inequality, or by multiplying or dividing both sides of the inequality by a non-zero constant.

Q: Can I have multiple solutions to a linear inequality?

A: Yes, you can have multiple solutions to a linear inequality. For example, the inequality x ≤ 5 has multiple solutions, including x = 0, x = 1, x = 2, x = 3, x = 4, and x = 5.

Q: Can I have no solution to a linear inequality?

A: Yes, you can have no solution to a linear inequality. For example, the inequality x > 5 has no solution, since there is no value of x that satisfies the inequality.

Conclusion

In this article, we answered some of the most frequently asked questions about linear inequalities. We hope this article has been helpful in understanding how to solve linear inequalities and how to graph them.

Tips and Tricks

  • When solving a linear inequality, make sure to isolate the variable on one side of the inequality.
  • When adding or subtracting a constant from both sides of an inequality, make sure to keep the direction of the inequality the same.
  • When multiplying or dividing both sides of an inequality by a non-zero constant, make sure to reverse the direction of the inequality if necessary.

Practice Problems

  1. Solve the inequality $2x + 5 \leq 17$.
  2. Solve the inequality $x - 3 \geq 2$.
  3. Solve the inequality $4x - 2 \leq 12$.

Solutions

  1. 2x+5≤172x + 5 \leq 17

    2x+5−5≤17−52x + 5 - 5 \leq 17 - 5

    2x≤122x \leq 12

    2x2≤122\frac{2x}{2} \leq \frac{12}{2}

    x≤6x \leq 6

  2. x−3≥2x - 3 \geq 2

    x−3+3≥2+3x - 3 + 3 \geq 2 + 3

    x≥5x \geq 5

  3. 4x−2≤124x - 2 \leq 12

    4x−2+2≤12+24x - 2 + 2 \leq 12 + 2

    4x≤144x \leq 14

    4x4≤144\frac{4x}{4} \leq \frac{14}{4}

    x≤3.5x \leq 3.5

Conclusion

In this article, we answered some of the most frequently asked questions about linear inequalities. We hope this article has been helpful in understanding how to solve linear inequalities and how to graph them.