Solve The Inequality:$\[ \frac{3}{10}(t+2)-\frac{t}{20} \geq \frac{3}{5} \\]Then, Consider The Possible Values Of \[$t\$\] From The Following Options:A. \[$t \geq 0\$\] B. \[$t \leq -\frac{24}{5}\$\] C. \[$1 \leq

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Introduction

Inequalities are mathematical expressions that compare two values, indicating whether one is greater than, less than, or equal to the other. Solving inequalities involves isolating the variable on one side of the inequality sign and determining the possible values of the variable that satisfy the inequality. In this article, we will focus on solving the inequality 310(t+2)−t20≥35\frac{3}{10}(t+2)-\frac{t}{20} \geq \frac{3}{5} and consider the possible values of tt from the given options.

Understanding the Inequality

The given inequality is 310(t+2)−t20≥35\frac{3}{10}(t+2)-\frac{t}{20} \geq \frac{3}{5}. To solve this inequality, we need to isolate the variable tt on one side of the inequality sign. We can start by simplifying the left-hand side of the inequality.

Simplifying the Left-Hand Side

To simplify the left-hand side, we can start by distributing the 310\frac{3}{10} to the terms inside the parentheses:

310(t+2)=3t10+610\frac{3}{10}(t+2) = \frac{3t}{10} + \frac{6}{10}

Now, we can rewrite the inequality as:

3t10+610−t20≥35\frac{3t}{10} + \frac{6}{10} - \frac{t}{20} \geq \frac{3}{5}

Combining Like Terms

Next, we can combine the like terms on the left-hand side:

3t10−t20+610≥35\frac{3t}{10} - \frac{t}{20} + \frac{6}{10} \geq \frac{3}{5}

To combine the fractions, we need to find a common denominator, which is 20:

6t20+1220≥35\frac{6t}{20} + \frac{12}{20} \geq \frac{3}{5}

Now, we can rewrite the inequality as:

6t+1220≥35\frac{6t + 12}{20} \geq \frac{3}{5}

Multiplying Both Sides

To eliminate the fractions, we can multiply both sides of the inequality by 20:

6t+12≥126t + 12 \geq 12

Isolating the Variable

Now, we can isolate the variable tt by subtracting 12 from both sides of the inequality:

6t≥06t \geq 0

Dividing Both Sides

Finally, we can divide both sides of the inequality by 6 to solve for tt:

t≥0t \geq 0

Evaluating the Possible Values of tt

Now that we have solved the inequality, we can evaluate the possible values of tt from the given options:

  • Option A: t≥0t \geq 0
  • Option B: t≤−245t \leq -\frac{24}{5}
  • Option C: 1≤t≤21 \leq t \leq 2

Based on our solution, we can see that the possible values of tt are t≥0t \geq 0. Therefore, the correct answer is Option A.

Conclusion

Solving inequalities involves isolating the variable on one side of the inequality sign and determining the possible values of the variable that satisfy the inequality. In this article, we solved the inequality 310(t+2)−t20≥35\frac{3}{10}(t+2)-\frac{t}{20} \geq \frac{3}{5} and evaluated the possible values of tt from the given options. We found that the possible values of tt are t≥0t \geq 0, which corresponds to Option A.

Frequently Asked Questions

Q: What is an inequality?

A: An inequality is a mathematical expression that compares two values, indicating whether one is greater than, less than, or equal to the other.

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 and determine the possible values of the variable that satisfy the inequality.

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

A: An equation is a mathematical expression that states that two values are equal, while an inequality is a mathematical expression that compares two values, indicating whether one is greater than, less than, or equal to the other.

Q: Can I use the same methods to solve inequalities as I would to solve equations?

A: No, the methods used to solve inequalities are different from those used to solve equations. Inequalities require a different set of rules and techniques to solve.

Final Thoughts

Solving inequalities is an essential skill in mathematics, and it requires a deep understanding of the concepts and techniques involved. By following the steps outlined in this article, you can solve inequalities and evaluate the possible values of the variable that satisfy the inequality. Remember to always isolate the variable on one side of the inequality sign and determine the possible values of the variable that satisfy the inequality. With practice and patience, you can become proficient in solving inequalities and apply this skill to a wide range of mathematical problems.

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Introduction

Solving inequalities is a crucial skill in mathematics, and it requires a deep understanding of the concepts and techniques involved. In this article, we will provide a Q&A guide to help you understand and solve inequalities. Whether you are a student, a teacher, or a math enthusiast, this guide will provide you with the knowledge and confidence to tackle inequalities with ease.

Q&A: Solving Inequalities

Q: What is an inequality?

A: An inequality is a mathematical expression that compares two values, indicating whether one is greater than, less than, or equal to the other.

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 and determine the possible values of the variable that satisfy the inequality.

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

A: An equation is a mathematical expression that states that two values are equal, while an inequality is a mathematical expression that compares two values, indicating whether one is greater than, less than, or equal to the other.

Q: Can I use the same methods to solve inequalities as I would to solve equations?

A: No, the methods used to solve inequalities are different from those used to solve equations. Inequalities require a different set of rules and techniques to solve.

Q: How do I isolate the variable in an inequality?

A: To isolate the variable, you need to perform operations on both sides of the inequality sign that will eliminate the constant term. This may involve adding, subtracting, multiplying, or dividing both sides of the inequality.

Q: What is the order of operations for solving inequalities?

A: The order of operations for solving inequalities is the same as for solving equations: parentheses, exponents, multiplication and division, and addition and subtraction.

Q: Can I use inverse operations to solve inequalities?

A: Yes, you can use inverse operations to solve inequalities. For example, if you have an inequality with a variable in the numerator, you can use the inverse operation of division to isolate the variable.

Q: How do I determine the possible values of the variable that satisfy the inequality?

A: To determine the possible values of the variable, you need to consider the direction of the inequality sign and the values of the constants involved. You may need to use a number line or a graph to visualize the solution.

Q: Can I use algebraic manipulations to solve inequalities?

A: Yes, you can use algebraic manipulations to solve inequalities. For example, you can use the distributive property to expand expressions and simplify the inequality.

Q: How do I check my solution to an inequality?

A: To check your solution, you need to substitute the value of the variable into the original inequality and verify that it is true.

Common Mistakes to Avoid

1. Not isolating the variable

Make sure to isolate the variable on one side of the inequality sign.

2. Not considering the direction of the inequality sign

Pay attention to the direction of the inequality sign and the values of the constants involved.

3. Not using inverse operations

Use inverse operations to isolate the variable and simplify the inequality.

4. Not checking the solution

Always check your solution to an inequality by substituting the value of the variable into the original inequality.

Conclusion

Solving inequalities is a crucial skill in mathematics, and it requires a deep understanding of the concepts and techniques involved. By following the steps outlined in this article and avoiding common mistakes, you can become proficient in solving inequalities and apply this skill to a wide range of mathematical problems. Remember to always isolate the variable on one side of the inequality sign, consider the direction of the inequality sign, and use inverse operations to simplify the inequality.

Frequently Asked Questions

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

A: A linear inequality is an inequality with a linear expression, while a quadratic inequality is an inequality with a quadratic expression.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you need to factor the quadratic expression and use the sign of the quadratic expression to determine the possible values of the variable.

Q: Can I use the quadratic formula to solve a quadratic inequality?

A: No, the quadratic formula is used to solve quadratic equations, not quadratic inequalities.

Q: How do I determine the possible values of the variable that satisfy a quadratic inequality?

A: To determine the possible values of the variable, you need to consider the sign of the quadratic expression and the values of the constants involved.

Final Thoughts

Solving inequalities is a challenging but rewarding skill in mathematics. By following the steps outlined in this article and avoiding common mistakes, you can become proficient in solving inequalities and apply this skill to a wide range of mathematical problems. Remember to always isolate the variable on one side of the inequality sign, consider the direction of the inequality sign, and use inverse operations to simplify the inequality. With practice and patience, you can become a master of solving inequalities and tackle even the most challenging problems with confidence.