1. Solve The Following Inequalities And Show The Graphs Of Their Solutions On A Number Line.a. $\[ 5x + 1 \ \textless \ 2x + 7 \\]b. $\[ \begin{cases} 3x + 1 \ \textless \ 16 \\ -2x + 5 \leq 13 \end{cases}

Introduction

Inequalities are mathematical statements that compare two expressions using greater than, less than, greater than or equal to, or less than or equal to. Solving inequalities involves finding the values of the variable that make the inequality true. In this article, we will solve two inequalities and show the graphs of their solutions on a number line.

Solving Inequality a

The first inequality is:

$$5x + 1 < 2x + 7$$

To solve this inequality, we need to isolate the variable x. We can do this by subtracting 2x from both sides of the inequality:

$$5x - 2x + 1 < 2x - 2x + 7$$

This simplifies to:

$$3x + 1 < 7$$

Next, we can subtract 1 from both sides of the inequality:

$$3x + 1 - 1 < 7 - 1$$

This simplifies to:

$$3x < 6$$

Finally, we can divide both sides of the inequality by 3:

$$\frac{3x}{3} < \frac{6}{3}$$

This simplifies to:

$$x < 2$$

The solution to the inequality is x < 2. This means that any value of x that is less than 2 will make the inequality true.

Graphing the Solution on a Number Line

To graph the solution on a number line, we need to draw a line that represents the inequality x < 2. We can do this by drawing a line that starts at negative infinity and ends at 2. We can also draw a small open circle at x = 2 to indicate that 2 is not included in the solution.

Solving Inequality b

The second inequality is:

$$\begin{cases} 3x + 1 < 16 \\ -2x + 5 \leq 13 \end{cases}$$

To solve this inequality, we need to solve each part of the inequality separately.

Solving the First Part of the Inequality

The first part of the inequality is:

$$3x + 1 < 16$$

To solve this inequality, we can subtract 1 from both sides of the inequality:

$$3x + 1 - 1 < 16 - 1$$

This simplifies to:

$$3x < 15$$

Next, we can divide both sides of the inequality by 3:

$$\frac{3x}{3} < \frac{15}{3}$$

This simplifies to:

$$x < 5$$

The solution to the first part of the inequality is x < 5.

Solving the Second Part of the Inequality

The second part of the inequality is:

$$-2x + 5 \leq 13$$

To solve this inequality, we can subtract 5 from both sides of the inequality:

$$-2x + 5 - 5 \leq 13 - 5$$

This simplifies to:

$$-2x \leq 8$$

Next, we can divide both sides of the inequality by -2. Remember that when we divide or multiply an inequality by a negative number, we need to reverse the direction of the inequality:

$$\frac{-2x}{-2} \geq \frac{8}{-2}$$

This simplifies to:

$$x \geq -4$$

The solution to the second part of the inequality is x ≥ -4.

Finding the Intersection of the Solutions

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

The solution to the first inequality is x < 5, and the solution to the second inequality is x ≥ -4. To find the intersection of these two solution sets, we need to find the values of x that satisfy both inequalities.

We can do this by drawing a number line and marking the values of x that satisfy both inequalities. The intersection of the two solution sets is the values of x that are greater than or equal to -4 and less than 5.

Graphing the Solution on a Number Line

To graph the solution on a number line, we need to draw a line that represents the intersection of the two solution sets. We can do this by drawing a line that starts at -4 and ends at 5. We can also draw a small closed circle at x = -4 to indicate that -4 is included in the solution, and a small open circle at x = 5 to indicate that 5 is not included in the solution.

Conclusion

Introduction

In the previous article, we solved two inequalities and showed the graphs of their solutions on a number line. In this article, we will answer some common questions that students often have when solving inequalities and graphing their solutions on a number line.

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 solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable x. You can do this by adding or subtracting the same value to both sides of the inequality, or by multiplying or dividing both sides of the inequality by the same non-zero value.

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

A: To graph a linear inequality on a number line, you need to draw a line that represents the inequality. If the inequality is of the form x < a, you draw a line that starts at negative infinity and ends at a. If the inequality is of the form x > a, you draw a line that starts at a and ends at positive infinity.

Q: What is the difference between a closed circle and an open circle on a number line?

A: A closed circle on a number line represents a value that is included in the solution set. An open circle on a number line represents a value that is not included in the solution set.

Q: How do I solve a system of linear inequalities?

A: To solve a system of linear inequalities, you need to find the values of x that satisfy both inequalities. You can do this by graphing the inequalities on a number line and finding the intersection of the two solution sets.

Q: What is the intersection of two solution sets?

A: The intersection of two solution sets is the set of values that are common to both solution sets. In other words, it is the set of values that satisfy both inequalities.

Q: How do I graph the intersection of two solution sets on a number line?

A: To graph the intersection of two solution sets on a number line, you need to draw a line that represents the intersection of the two solution sets. You can do this by drawing a line that starts at the smallest value in the intersection and ends at the largest value in the intersection.

Q: What are some common mistakes to avoid when solving inequalities and graphing their solutions on a number line?

A: Some common mistakes to avoid when solving inequalities and graphing their solutions on a number line include:

• Not isolating the variable x
• Not drawing the correct line on the number line
• Not including or excluding the correct values in the solution set
• Not finding the intersection of the two solution sets

Conclusion

In this article, we answered some common questions that students often have when solving inequalities and graphing their solutions on a number line. We also discussed some common mistakes to avoid when solving inequalities and graphing their solutions on a number line. By following these tips and avoiding these common mistakes, you can become more confident and proficient in solving inequalities and graphing their solutions on a number line.

Additional Resources

If you are struggling with solving inequalities and graphing their solutions on a number line, there are many additional resources available to help you. Some of these resources include:

• Online tutorials and videos
• Practice problems and worksheets
• Study guides and textbooks
• Online communities and forums

By taking advantage of these resources, you can get the help and support you need to succeed in solving inequalities and graphing their solutions on a number line.