Which Best Describes The Solution Set For The Inequality Below?$3x + 7 \leq 4x - 8$ Or $-2x + 3 \geq 1$A. $x \leq 1$ Or $x \geq 15$ B. $x \geq 1$ Or $x \leq 15$ C. $x \geq 1$ D. $x

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In mathematics, inequalities are a fundamental concept that deals with the comparison of two or more expressions. Solving inequalities involves finding the values of the variable that satisfy the given inequality. In this article, we will focus on solving two given inequalities and determining the solution set for each.

Understanding the Basics of Inequalities

Inequalities are mathematical statements that compare two or more expressions using greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤) symbols. The goal of solving an inequality is to isolate the variable and determine the values that satisfy the given inequality.

Types of Inequalities

There are two main types of inequalities: linear and nonlinear. Linear inequalities involve a linear expression, while nonlinear inequalities involve a nonlinear expression.

Solving Linear Inequalities

Linear inequalities can be solved using basic algebraic operations such as addition, subtraction, multiplication, and division. The goal is to isolate the variable and determine the values that satisfy the given inequality.

Solving the First Inequality: 3x+7≤4x−83x + 7 \leq 4x - 8

The first inequality is 3x+7≤4x−83x + 7 \leq 4x - 8. To solve this inequality, we need to isolate the variable xx.

Step 1: Subtract 3x3x from both sides

Subtracting 3x3x from both sides of the inequality gives us:

7≤x−87 \leq x - 8

Step 2: Add 8 to both sides

Adding 8 to both sides of the inequality gives us:

15≤x15 \leq x

Step 3: Write the solution in interval notation

The solution to the inequality 3x+7≤4x−83x + 7 \leq 4x - 8 is x≥15x \geq 15.

Solving the Second Inequality: −2x+3≥1-2x + 3 \geq 1

The second inequality is −2x+3≥1-2x + 3 \geq 1. To solve this inequality, we need to isolate the variable xx.

Step 1: Subtract 3 from both sides

Subtracting 3 from both sides of the inequality gives us:

−2x≥−2-2x \geq -2

Step 2: Divide both sides by -2

Dividing both sides of the inequality by -2 gives us:

x≤1x \leq 1

Step 3: Write the solution in interval notation

The solution to the inequality −2x+3≥1-2x + 3 \geq 1 is x≤1x \leq 1.

Determining the Solution Set

Now that we have solved both inequalities, we need to determine the solution set for each. The solution set is the set of all values that satisfy the given inequality.

Solution Set for the First Inequality

The solution set for the inequality 3x+7≤4x−83x + 7 \leq 4x - 8 is x≥15x \geq 15.

Solution Set for the Second Inequality

The solution set for the inequality −2x+3≥1-2x + 3 \geq 1 is x≤1x \leq 1.

Conclusion

In conclusion, solving inequalities involves finding the values of the variable that satisfy the given inequality. We have solved two given inequalities and determined the solution set for each. The solution set for the first inequality is x≥15x \geq 15, and the solution set for the second inequality is x≤1x \leq 1.

Final Answer

The final answer is:

A. x≤1x \leq 1 or x≥15x \geq 15

This is the correct solution set for the given inequalities.

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In the previous article, we discussed solving inequalities and determining the solution set for each. In this article, we will address some frequently asked questions (FAQs) on solving inequalities.

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

A: A linear inequality involves a linear expression, while a nonlinear inequality involves a nonlinear expression. Linear inequalities can be solved using basic algebraic operations, while nonlinear inequalities require more advanced techniques.

Q: How do I solve an inequality with fractions?

A: To solve an inequality with fractions, you need to eliminate the fractions by multiplying both sides of the inequality by the least common multiple (LCM) of the denominators.

Q: Can I add or subtract the same value to both sides of an inequality?

A: Yes, you can add or subtract the same value to both sides of an inequality. However, you cannot multiply or divide both sides of an inequality by a negative value, as this would change the direction of the inequality.

Q: How do I determine the solution set for an inequality?

A: To determine the solution set for an inequality, you need to isolate the variable and determine the values that satisfy the given inequality. The solution set is the set of all values that satisfy the inequality.

Q: Can I use a calculator to solve an inequality?

A: Yes, you can use a calculator to solve an inequality. However, you need to be careful when using a calculator, as it may not always give you the correct solution.

Q: How do I graph an inequality on a number line?

A: To graph an inequality on a number line, you need to plot a point on the number line that satisfies the inequality and then shade the region to the left or right of the point, depending on the direction of the inequality.

Q: Can I solve an inequality with absolute value?

A: Yes, you can solve an inequality with absolute value. To solve an inequality with absolute value, you need to consider two cases: one where the expression inside the absolute value is positive, and one where the expression inside the absolute value is negative.

Q: How do I solve an inequality with multiple variables?

A: To solve an inequality with multiple variables, you need to isolate one variable and then substitute the expression for that variable into the inequality. You can then solve the resulting inequality for the remaining variables.

Q: Can I use algebraic properties to solve an inequality?

A: Yes, you can use algebraic properties to solve an inequality. For example, you can use the commutative, associative, and distributive properties to simplify the inequality and make it easier to solve.

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

A: To check your solution to an inequality, you need to plug the solution back into the original inequality and verify that it is true. If the solution is not true, then you need to re-solve the inequality.

Q: Can I use technology to solve an inequality?

A: Yes, you can use technology to solve an inequality. For example, you can use a graphing calculator or a computer algebra system to solve an inequality.

Conclusion

In conclusion, solving inequalities involves finding the values of the variable that satisfy the given inequality. We have addressed some frequently asked questions (FAQs) on solving inequalities and provided tips and techniques for solving inequalities.

Final Answer

The final answer is:

  • Q: What is the difference between a linear inequality and a nonlinear inequality? A: A linear inequality involves a linear expression, while a nonlinear inequality involves a nonlinear expression.
  • Q: How do I solve an inequality with fractions? A: To solve an inequality with fractions, you need to eliminate the fractions by multiplying both sides of the inequality by the least common multiple (LCM) of the denominators.
  • Q: Can I add or subtract the same value to both sides of an inequality? A: Yes, you can add or subtract the same value to both sides of an inequality.
  • Q: How do I determine the solution set for an inequality? A: To determine the solution set for an inequality, you need to isolate the variable and determine the values that satisfy the given inequality.
  • Q: Can I use a calculator to solve an inequality? A: Yes, you can use a calculator to solve an inequality.
  • Q: How do I graph an inequality on a number line? A: To graph an inequality on a number line, you need to plot a point on the number line that satisfies the inequality and then shade the region to the left or right of the point, depending on the direction of the inequality.
  • Q: Can I solve an inequality with absolute value? A: Yes, you can solve an inequality with absolute value.
  • Q: How do I solve an inequality with multiple variables? A: To solve an inequality with multiple variables, you need to isolate one variable and then substitute the expression for that variable into the inequality.
  • Q: Can I use algebraic properties to solve an inequality? A: Yes, you can use algebraic properties to solve an inequality.
  • Q: How do I check my solution to an inequality? A: To check your solution to an inequality, you need to plug the solution back into the original inequality and verify that it is true.
  • Q: Can I use technology to solve an inequality? A: Yes, you can use technology to solve an inequality.