Solve The Inequality 7 8 Y ≥ − 6 \frac{7}{8} Y \geq -6 8 7 ​ Y ≥ − 6 And Express The Solution In Interval Notation. Solution: Y ∈ [ − 6.857 , ∞ Y \in [-6.857, \infty Y ∈ [ − 6.857 , ∞ ].

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Introduction

Inequalities are mathematical expressions that compare two values using greater than, less than, greater than or equal to, or less than or equal to. Solving inequalities involves isolating the variable on one side of the inequality sign. In this article, we will focus on solving the inequality 78y≥−6\frac{7}{8} y \geq -6 and express the solution in interval notation.

Understanding the Inequality

The given inequality is 78y≥−6\frac{7}{8} y \geq -6. To solve this inequality, we need to isolate the variable yy on one side of the inequality sign. The first step is to multiply both sides of the inequality by the reciprocal of 78\frac{7}{8}, which is 87\frac{8}{7}.

Multiplying Both Sides

Multiplying both sides of the inequality by 87\frac{8}{7} gives us:

y≥87⋅(−6)y \geq \frac{8}{7} \cdot (-6)

Simplifying the Expression

To simplify the expression, we multiply 87\frac{8}{7} by −6-6:

y≥−487y \geq -\frac{48}{7}

Expressing the Solution in Interval Notation

The solution to the inequality is all values of yy that are greater than or equal to −487-\frac{48}{7}. To express this in interval notation, we use the following notation:

y∈[−487,∞)y \in \left[-\frac{48}{7}, \infty\right)

Why Interval Notation?

Interval notation is a way of expressing a set of numbers using square brackets or parentheses. In this case, we use a left square bracket to indicate that the value −487-\frac{48}{7} is included in the solution set, and a right parenthesis to indicate that all values greater than −487-\frac{48}{7} are also included.

Real-World Applications

Solving inequalities has many real-world applications. For example, in finance, inequalities can be used to model the growth of investments over time. In engineering, inequalities can be used to design and optimize systems. In medicine, inequalities can be used to model the spread of diseases.

Conclusion

Solving inequalities is an important skill in mathematics. By following the steps outlined in this article, we can solve inequalities and express the solution in interval notation. Remember to always multiply both sides of the inequality by the reciprocal of the coefficient of the variable, and to simplify the expression by multiplying the numbers. With practice, you will become proficient in solving inequalities and applying them to real-world problems.

Common Mistakes to Avoid

When solving inequalities, there are several common mistakes to avoid. These include:

  • Not multiplying both sides of the inequality by the reciprocal of the coefficient of the variable
  • Not simplifying the expression by multiplying the numbers
  • Not using interval notation to express the solution

Tips and Tricks

Here are some tips and tricks to help you solve inequalities:

  • Use a calculator to check your work
  • Read the problem carefully and identify the inequality sign
  • Use a diagram to visualize the solution set

Practice Problems

Here are some practice problems to help you practice solving inequalities:

  • Solve the inequality 2x≥52x \geq 5 and express the solution in interval notation.
  • Solve the inequality x−3≥2x - 3 \geq 2 and express the solution in interval notation.
  • Solve the inequality 12x≤3\frac{1}{2} x \leq 3 and express the solution in interval notation.

Answer Key

Here are the answers to the practice problems:

  • x∈[2.5,∞)x \in [2.5, \infty)
  • x∈[5,∞)x \in [5, \infty)
  • x∈[−6,∞)x \in [-6, \infty)

Conclusion

Introduction

In our previous article, we discussed how to solve inequalities and express the solution in interval notation. In this article, we will provide a Q&A guide to help you better understand the concepts and techniques involved in solving inequalities.

Q: What is an inequality?

A: An inequality is a mathematical expression that compares two values using greater than, less than, greater than or equal to, or less than or equal to.

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable on one side of the inequality sign. This involves multiplying both sides of the inequality by the reciprocal of the coefficient of the variable, and simplifying the expression by multiplying the numbers.

Q: What is the reciprocal of a fraction?

A: The reciprocal of a fraction is obtained by swapping the numerator and denominator. For example, the reciprocal of 12\frac{1}{2} is 22.

Q: How do I multiply both sides of an inequality by a fraction?

A: To multiply both sides of an inequality by a fraction, you need to multiply both sides by the reciprocal of the fraction. For example, to multiply both sides of the inequality 2x≥52x \geq 5 by 12\frac{1}{2}, you need to multiply both sides by 22, which gives you x≥52x \geq \frac{5}{2}.

Q: What is interval notation?

A: Interval notation is a way of expressing a set of numbers using square brackets or parentheses. In this case, we use a left square bracket to indicate that the value is included in the solution set, and a right parenthesis to indicate that all values greater than the value are also included.

Q: How do I express the solution to an inequality in interval notation?

A: To express the solution to an inequality in interval notation, you need to identify the value that is included in the solution set, and the values that are greater than or less than the value. For example, the solution to the inequality x≥2x \geq 2 is expressed in interval notation as [2,∞)[2, \infty).

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

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

  • Not multiplying both sides of the inequality by the reciprocal of the coefficient of the variable
  • Not simplifying the expression by multiplying the numbers
  • Not using interval notation to express the solution

Q: How can I practice solving inequalities?

A: You can practice solving inequalities by working through practice problems, such as the ones provided in our previous article. You can also try solving inequalities on your own, using different types of inequalities and variables.

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

A: Solving inequalities has many real-world applications, including:

  • Finance: Inequalities can be used to model the growth of investments over time.
  • Engineering: Inequalities can be used to design and optimize systems.
  • Medicine: Inequalities can be used to model the spread of diseases.

Conclusion

Solving inequalities is an important skill in mathematics. By following the steps outlined in this article, you can solve inequalities and express the solution in interval notation. Remember to always multiply both sides of the inequality by the reciprocal of the coefficient of the variable, and to simplify the expression by multiplying the numbers. With practice, you will become proficient in solving inequalities and applying them to real-world problems.

Additional Resources

For more information on solving inequalities, you can consult the following resources:

  • Textbooks: "Algebra and Trigonometry" by Michael Sullivan, "College Algebra" by James Stewart
  • Online resources: Khan Academy, Mathway, Wolfram Alpha
  • Practice problems: MIT OpenCourseWare, Math Open Reference

Final Tips

Here are some final tips to help you succeed in solving inequalities:

  • Practice regularly: Practice solving inequalities regularly to build your skills and confidence.
  • Use online resources: Use online resources, such as Khan Academy and Mathway, to supplement your learning.
  • Seek help: Don't be afraid to seek help if you're struggling with a particular concept or problem.