Which Solutions Do The Inequalities $\frac{1}{2}(5x + 3) \geq 14$ And $-2y + 6 \geq -8$ Have In Common?Show Your Work.

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Introduction

Inequalities are mathematical expressions that compare two values, often with a greater-than or less-than symbol. Solving inequalities involves finding the values of the variable that make the inequality true. In this article, we will explore the solutions to two inequalities, 12(5x+3)≥14\frac{1}{2}(5x + 3) \geq 14 and −2y+6≥−8-2y + 6 \geq -8, and determine which solutions they have in common.

Solving the First Inequality

The first inequality is 12(5x+3)≥14\frac{1}{2}(5x + 3) \geq 14. To solve this inequality, we need to isolate the variable xx. We can start by multiplying both sides of the inequality by 2 to eliminate the fraction.

$\frac{1}{2}(5x + 3) \geq 14$
Multiplying both sides by 2:
$5x + 3 \geq 28$

Next, we can subtract 3 from both sides of the inequality to isolate the term with the variable.

$5x + 3 \geq 28$
Subtracting 3 from both sides:
$5x \geq 25$

Finally, we can divide both sides of the inequality by 5 to solve for xx.

$5x \geq 25$
Dividing both sides by 5:
$x \geq 5$

Solving the Second Inequality

The second inequality is −2y+6≥−8-2y + 6 \geq -8. To solve this inequality, we need to isolate the variable yy. We can start by subtracting 6 from both sides of the inequality to eliminate the constant term.

$-2y + 6 \geq -8$
Subtracting 6 from both sides:
$-2y \geq -14$

Next, we can divide both sides of the inequality by -2 to solve for yy. When we divide by a negative number, we need to reverse the direction of the inequality.

$-2y \geq -14$
Dividing both sides by -2:
$y \leq 7$

Finding the Common Solutions

Now that we have solved both inequalities, we need to find the values of xx and yy that satisfy both inequalities. We can start by listing the solutions to each inequality.

  • For the first inequality, x≥5x \geq 5.
  • For the second inequality, y≤7y \leq 7.

To find the common solutions, we need to find the values of xx and yy that satisfy both inequalities. We can do this by finding the intersection of the two solution sets.

$x \geq 5$
$y \leq 7$

The common solutions are the values of xx and yy that satisfy both inequalities. In this case, the common solutions are all values of xx that are greater than or equal to 5, and all values of yy that are less than or equal to 7.

Conclusion

In this article, we have solved two inequalities, 12(5x+3)≥14\frac{1}{2}(5x + 3) \geq 14 and −2y+6≥−8-2y + 6 \geq -8, and determined which solutions they have in common. We found that the common solutions are all values of xx that are greater than or equal to 5, and all values of yy that are less than or equal to 7. This shows that the solutions to the two inequalities have a common region of overlap.

Final Answer

The final answer is: 5≤x≤∞,−∞≤y≤7\boxed{5 \leq x \leq \infty, -\infty \leq y \leq 7}

Introduction

In our previous article, we explored the solutions to two inequalities, 12(5x+3)≥14\frac{1}{2}(5x + 3) \geq 14 and −2y+6≥−8-2y + 6 \geq -8, and determined which solutions they have in common. In this article, we will answer some frequently asked questions about solving inequalities and finding common solutions.

Q: What is the difference between solving an equation and solving an inequality?

A: Solving an equation involves finding the value of the variable that makes the equation true, whereas solving an inequality involves finding the values of the variable that make the inequality true.

Q: How do I know which direction to flip the inequality sign when dividing by a negative number?

A: When dividing by a negative number, you need to reverse the direction of the inequality sign. For example, if you have the inequality x≥5x \geq 5 and you divide both sides by -2, the resulting inequality would be x≤−52x \leq -\frac{5}{2}.

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

A: While some of the steps are similar, there are some key differences. For example, when solving an inequality, you need to consider the direction of the inequality sign and whether you need to flip it when dividing by a negative number.

Q: How do I find the common solutions to two inequalities?

A: To find the common solutions to two inequalities, you need to find the intersection of the two solution sets. This involves finding the values of the variable that satisfy both inequalities.

Q: Can I use a graph to help me find the common solutions to two inequalities?

A: Yes, you can use a graph to help you find the common solutions to two inequalities. By graphing the two inequalities on the same coordinate plane, you can visually identify the region of overlap, which represents the common solutions.

Q: What if I have a system of linear inequalities? How do I find the common solutions?

A: If you have a system of linear inequalities, you can use a similar approach to find the common solutions. You can graph the inequalities on the same coordinate plane and identify the region of overlap, which represents the common solutions.

Q: Can I use technology to help me solve inequalities and find common solutions?

A: Yes, you can use technology, such as graphing calculators or computer software, to help you solve inequalities and find common solutions. These tools can help you visualize the solution sets and identify the common solutions.

Conclusion

In this article, we have answered some frequently asked questions about solving inequalities and finding common solutions. We hope that this article has provided you with a better understanding of how to approach these types of problems and has given you the confidence to tackle more complex inequalities.

Final Answer

The final answer is: There is no final numerical answer to this article, as it is a Q&A article. However, we hope that the information provided has been helpful and informative.