Given The System Of Inequalities:$\[ \begin{array}{l} 4x - 5y \leq 1 \\ \frac{1}{2}y - X \leq 3 \end{array} \\]Which Shows The Given Inequalities In Slope-intercept Form?A. \[$y \leq \frac{4}{5}x - \frac{1}{5}\$\]B. \[$y \leq 2x +

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Introduction

Solving systems of inequalities is a crucial concept in mathematics, particularly in algebra and geometry. It involves finding the solution set that satisfies multiple inequalities simultaneously. In this article, we will focus on solving a system of two linear inequalities and converting them into slope-intercept form.

Understanding the System of Inequalities

The given system of inequalities is:

{ \begin{array}{l} 4x - 5y \leq 1 \\ \frac{1}{2}y - x \leq 3 \end{array} \}

To solve this system, we need to find the solution set that satisfies both inequalities simultaneously.

Converting the First Inequality to Slope-Intercept Form

The first inequality is 4x−5y≤14x - 5y \leq 1. To convert it into slope-intercept form, we need to isolate yy on one side of the inequality.

Step 1: Subtract 4x4x from both sides of the inequality.

−5y≤−4x+1-5y \leq -4x + 1

Step 2: Divide both sides of the inequality by −5-5.

y≥45x−15y \geq \frac{4}{5}x - \frac{1}{5}

However, since we are dividing by a negative number, the inequality sign is flipped.

Converting the Second Inequality to Slope-Intercept Form

The second inequality is 12y−x≤3\frac{1}{2}y - x \leq 3. To convert it into slope-intercept form, we need to isolate yy on one side of the inequality.

Step 1: Add xx to both sides of the inequality.

12y≤x+3\frac{1}{2}y \leq x + 3

Step 2: Multiply both sides of the inequality by 22.

y≤2x+6y \leq 2x + 6

Comparing the Options

Now that we have converted both inequalities into slope-intercept form, we can compare them with the given options.

Option A: y≤45x−15y \leq \frac{4}{5}x - \frac{1}{5}

Option B: y≤2x+6y \leq 2x + 6

Based on our calculations, we can see that Option B is the correct answer.

Conclusion

Solving systems of inequalities requires careful manipulation of the inequalities to isolate the variable. By converting the inequalities into slope-intercept form, we can easily compare them with the given options. In this article, we have demonstrated how to solve a system of two linear inequalities and convert them into slope-intercept form.

Tips and Tricks

  • When solving systems of inequalities, it's essential to carefully manipulate the inequalities to isolate the variable.
  • Converting inequalities into slope-intercept form can help simplify the solution process.
  • When dividing by a negative number, the inequality sign is flipped.

Real-World Applications

Solving systems of inequalities has numerous real-world applications, including:

  • Finance: Solving systems of inequalities can help investors make informed decisions about investments and risk management.
  • Engineering: Solving systems of inequalities can help engineers design and optimize systems, such as electrical circuits and mechanical systems.
  • Computer Science: Solving systems of inequalities can help computer scientists develop algorithms and models for solving complex problems.

Practice Problems

Try solving the following system of inequalities:

{ \begin{array}{l} 2x + 3y \leq 5 \\ x - 2y \leq 1 \end{array} \}

Converting the inequalities into slope-intercept form, we get:

y≤−23x+53y \leq -\frac{2}{3}x + \frac{5}{3}

y≤12x+12y \leq \frac{1}{2}x + \frac{1}{2}

Which option is correct?

A. y≤−23x+53y \leq -\frac{2}{3}x + \frac{5}{3}

B. y≤12x+12y \leq \frac{1}{2}x + \frac{1}{2}

C. y≤−23x+53y \leq -\frac{2}{3}x + \frac{5}{3} and y≤12x+12y \leq \frac{1}{2}x + \frac{1}{2}

D. y≤−23x+53y \leq -\frac{2}{3}x + \frac{5}{3} or y≤12x+12y \leq \frac{1}{2}x + \frac{1}{2}

Answer: C

Introduction

Solving systems of inequalities is a crucial concept in mathematics, particularly in algebra and geometry. In our previous article, we discussed how to solve a system of two linear inequalities and convert them into slope-intercept form. In this article, we will provide a Q&A guide to help you better understand the concept and solve systems of inequalities.

Q&A

Q1: What is a system of inequalities?

A system of inequalities is a set of two or more inequalities that are related to each other. It involves finding the solution set that satisfies multiple inequalities simultaneously.

Q2: How do I solve a system of inequalities?

To solve a system of inequalities, you need to find the solution set that satisfies both inequalities simultaneously. You can do this by converting the inequalities into slope-intercept form and then comparing them.

Q3: What is slope-intercept form?

Slope-intercept form is a way of writing an inequality in the form y≤mx+by \leq mx + b, where mm is the slope and bb is the y-intercept.

Q4: How do I convert an inequality into slope-intercept form?

To convert an inequality into slope-intercept form, you need to isolate yy on one side of the inequality. You can do this by adding or subtracting the same value from both sides of the inequality.

Q5: What is the difference between a system of linear inequalities and a system of nonlinear inequalities?

A system of linear inequalities involves linear inequalities, while a system of nonlinear inequalities involves nonlinear inequalities. Nonlinear inequalities are more complex and require different techniques to solve.

Q6: Can I use the same techniques to solve a system of inequalities with more than two inequalities?

Yes, you can use the same techniques to solve a system of inequalities with more than two inequalities. However, the solution set may be more complex and require additional techniques to solve.

Q7: How do I graph a system of inequalities?

To graph a system of inequalities, you need to graph each inequality separately and then find the intersection of the two graphs.

Q8: What is the importance of solving systems of inequalities?

Solving systems of inequalities has numerous real-world applications, including finance, engineering, and computer science. It helps us make informed decisions and optimize systems.

Q9: Can I use technology to solve systems of inequalities?

Yes, you can use technology, such as graphing calculators or computer software, to solve systems of inequalities. However, it's essential to understand the underlying concepts and techniques to use technology effectively.

Q10: How do I practice solving systems of inequalities?

To practice solving systems of inequalities, you can try solving problems on your own or use online resources, such as worksheets or video tutorials.

Conclusion

Solving systems of inequalities requires careful manipulation of the inequalities to isolate the variable. By converting the inequalities into slope-intercept form and comparing them, you can find the solution set that satisfies both inequalities simultaneously. In this article, we have provided a Q&A guide to help you better understand the concept and solve systems of inequalities.

Tips and Tricks

  • When solving systems of inequalities, it's essential to carefully manipulate the inequalities to isolate the variable.
  • Converting inequalities into slope-intercept form can help simplify the solution process.
  • When dividing by a negative number, the inequality sign is flipped.

Real-World Applications

Solving systems of inequalities has numerous real-world applications, including:

  • Finance: Solving systems of inequalities can help investors make informed decisions about investments and risk management.
  • Engineering: Solving systems of inequalities can help engineers design and optimize systems, such as electrical circuits and mechanical systems.
  • Computer Science: Solving systems of inequalities can help computer scientists develop algorithms and models for solving complex problems.

Practice Problems

Try solving the following system of inequalities:

{ \begin{array}{l} x + 2y \leq 3 \\ 2x - 3y \leq 5 \end{array} \}

Converting the inequalities into slope-intercept form, we get:

y≤−12x+32y \leq -\frac{1}{2}x + \frac{3}{2}

y≤23x−53y \leq \frac{2}{3}x - \frac{5}{3}

Which option is correct?

A. y≤−12x+32y \leq -\frac{1}{2}x + \frac{3}{2}

B. y≤23x−53y \leq \frac{2}{3}x - \frac{5}{3}

C. y≤−12x+32y \leq -\frac{1}{2}x + \frac{3}{2} and y≤23x−53y \leq \frac{2}{3}x - \frac{5}{3}

D. y≤−12x+32y \leq -\frac{1}{2}x + \frac{3}{2} or y≤23x−53y \leq \frac{2}{3}x - \frac{5}{3}

Answer: C