Question 5The Following Inequality Is Given: $3(x+7)\ \textless \ \frac{x}{2}+1$5.1 Solve For $x$ In The Inequality.5.2 Represent Your Answer To Question 5.1 On A Number Line.

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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. In this article, we will focus on solving the given inequality: $3(x+7)\ \textless \ \frac{x}{2}+1$

Step 1: Distribute and Simplify

To begin solving the inequality, we need to distribute the 3 to the terms inside the parentheses and simplify the expression.

3(x+7)=3x+213(x+7) = 3x + 21

Now, the inequality becomes:

3x+21 \textless x2+13x + 21\ \textless \ \frac{x}{2}+1

Step 2: Subtract 21 from Both Sides

Next, we need to isolate the variable x by subtracting 21 from both sides of the inequality.

3x+21−21 \textless x2+1−213x + 21 - 21\ \textless \ \frac{x}{2}+1 - 21

This simplifies to:

3x \textless x2−203x\ \textless \ \frac{x}{2}-20

Step 3: Multiply Both Sides by 2

To eliminate the fraction, we need to multiply both sides of the inequality by 2.

2(3x) \textless 2(x2−20)2(3x)\ \textless \ 2(\frac{x}{2}-20)

This simplifies to:

6x \textless x−406x\ \textless \ x-40

Step 4: Subtract x from Both Sides

Next, we need to isolate the variable x by subtracting x from both sides of the inequality.

6x−x \textless x−40−x6x - x\ \textless \ x-40 - x

This simplifies to:

5x \textless −405x\ \textless \ -40

Step 5: Divide Both Sides by 5

Finally, we need to isolate the variable x by dividing both sides of the inequality by 5.

5x5 \textless −405\frac{5x}{5}\ \textless \ \frac{-40}{5}

This simplifies to:

x \textless −8x\ \textless \ -8

Representing the Answer on a Number Line

To represent the answer on a number line, we need to draw a line with the numbers on it and shade the region that satisfies the inequality.

The number line will have the following format:

| | -10 | -9 | -8 | -7 | -6 | -5 | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

The region that satisfies the inequality will be shaded to the left of -8.

Conclusion

In this article, we solved the given inequality $3(x+7)\ \textless \ \frac{x}{2}+1$ and represented the answer on a number line. We used the steps of distributing and simplifying, subtracting 21 from both sides, multiplying both sides by 2, subtracting x from both sides, and dividing both sides by 5 to isolate the variable x. The final answer is x < -8.

Key Takeaways

  • To solve an inequality, we need to isolate the variable on one side of the inequality sign.
  • We can use the steps of distributing and simplifying, subtracting, multiplying, and dividing to isolate the variable.
  • The final answer should be represented on a number line to show the region that satisfies the inequality.

Frequently Asked Questions

  • What is the final answer to the inequality $3(x+7)\ \textless \ \frac{x}{2}+1$?
    • The final answer is x < -8.
  • How do I represent the answer on a number line?
    • To represent the answer on a number line, we need to draw a line with the numbers on it and shade the region that satisfies the inequality.
  • What are the steps to solve an inequality?
    • The steps to solve an inequality are distributing and simplifying, subtracting, multiplying, and dividing to isolate the variable.
      Frequently Asked Questions: 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. You can use the steps of distributing and simplifying, subtracting, multiplying, and dividing to isolate the variable.

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

A: A linear inequality is an inequality that can be written in the form ax + b < c, where a, b, and c are constants. A quadratic inequality is an inequality that can be written in the form ax^2 + bx + c < 0, where a, b, and c are constants.

Q: How do I represent the solution to an inequality on a number line?

A: To represent the solution to an inequality on a number line, you need to draw a line with the numbers on it and shade the region that satisfies the inequality.

Q: What is the concept of the "solution set" in solving inequalities?

A: The solution set is the set of all values of the variable that satisfy the inequality. It is the region on the number line that is shaded to represent the solution to the inequality.

Q: Can I use the same steps to solve a system of inequalities as I would to solve a system of equations?

A: No, you cannot use the same steps to solve a system of inequalities as you would to solve a system of equations. Solving a system of inequalities requires a different approach, such as graphing the inequalities on a number line and finding the intersection of the shaded regions.

Q: How do I determine the direction of the inequality sign when solving a system of inequalities?

A: When solving a system of inequalities, you need to determine the direction of the inequality sign by considering the signs of the coefficients of the variables. If the coefficient of a variable is positive, the inequality sign should point in the positive direction. If the coefficient of a variable is negative, the inequality sign should point in the negative direction.

Q: Can I use algebraic methods to solve a system of inequalities?

A: Yes, you can use algebraic methods to solve a system of inequalities. However, these methods may not always be the most efficient or effective way to solve the system.

Q: What is the importance of understanding the concept of inequalities in real-world applications?

A: Understanding the concept of inequalities is crucial in real-world applications, such as finance, economics, and engineering. Inequalities are used to model and analyze complex systems, make predictions, and make informed decisions.

Q: Can I use technology, such as graphing calculators or computer software, to solve inequalities?

A: Yes, you can use technology, such as graphing calculators or computer software, to solve inequalities. These tools can help you visualize the solution to the inequality and make it easier to understand the concept.

Q: How do I choose the best method to solve an inequality?

A: To choose the best method to solve an inequality, you need to consider the complexity of the inequality, the number of variables involved, and the level of precision required. You may need to use a combination of algebraic and graphical methods to solve the inequality.

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

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

  • Not following the order of operations
  • Not considering the signs of the coefficients of the variables
  • Not using the correct inequality sign
  • Not checking the solution for extraneous solutions

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

A: To check your solution to an inequality for extraneous solutions, you need to plug the solution back into the original inequality and verify that it is true. If the solution is not true, it is an extraneous solution and should be discarded.