Representing the Solution Set on a Number Line

Oct 13, 2025 by recoveryshake.com 34 views

In mathematics, inequalities are used to compare two or more values. They are an essential part of algebra and are used to solve a wide range of problems. In this article, we will focus on solving three different types of inequalities and representing their solution sets on a number line.

Solving Inequality a) $2(x+3) \ \textless \ 10$

To solve the inequality $2(x+3) \ \textless \ 10$ , we need to follow the order of operations (PEMDAS) and isolate the variable x.

Step 1: Distribute the 2 to the terms inside the parentheses

$2(x+3) = 2x + 6$

Step 2: Rewrite the inequality with the distributed terms

$2x + 6 \ \textless \ 10$

Step 3: Subtract 6 from both sides of the inequality

$2x \ \textless \ 4$

Step 4: Divide both sides of the inequality by 2

$x \ \textless \ 2$

The solution to the inequality $2(x+3) \ \textless \ 10$ is $x \ \textless \ 2$ . This means that any value of x that is less than 2 will satisfy the inequality.

Representing the Solution Set on a Number Line

To represent the solution set on a number line, we draw a line and mark the point x = 2 with a closed circle, indicating that it is not included in the solution set. We then shade the region to the left of the point x = 2, indicating that all values of x less than 2 are part of the solution set.

Solving Inequality b) $3t - 4 \geq t + 6$

To solve the inequality $3t - 4 \geq t + 6$ , we need to follow the order of operations (PEMDAS) and isolate the variable t.

Step 1: Add 4 to both sides of the inequality

$3t \geq t + 10$

Step 2: Subtract t from both sides of the inequality

$2t \geq 10$

Step 3: Divide both sides of the inequality by 2

$t \geq 5$

The solution to the inequality $3t - 4 \geq t + 6$ is $t \geq 5$ . This means that any value of t that is greater than or equal to 5 will satisfy the inequality.

Representing the Solution Set on a Number Line

To represent the solution set on a number line, we draw a line and mark the point t = 5 with a closed circle, indicating that it is included in the solution set. We then shade the region to the right of the point t = 5, indicating that all values of t greater than or equal to 5 are part of the solution set.

Solving Inequality c) $x + 12 \leq 30 - 2x$

To solve the inequality $x + 12 \leq 30 - 2x$ , we need to follow the order of operations (PEMDAS) and isolate the variable x.

Step 1: Add 2x to both sides of the inequality

$3x + 12 \leq 30$

Step 2: Subtract 12 from both sides of the inequality

$3x \leq 18$

Step 3: Divide both sides of the inequality by 3

$x \leq 6$

The solution to the inequality $x + 12 \leq 30 - 2x$ is $x \leq 6$ . This means that any value of x that is less than or equal to 6 will satisfy the inequality.

Representing the Solution Set on a Number Line

To represent the solution set on a number line, we draw a line and mark the point x = 6 with a closed circle, indicating that it is included in the solution set. We then shade the region to the left of the point x = 6, indicating that all values of x less than or equal to 6 are part of the solution set.

Conclusion

In this article, we solved three different types of inequalities and represented their solution sets on a number line. We used the order of operations (PEMDAS) and isolated the variable in each inequality. We then represented the solution set on a number line by drawing a line and marking the point with a closed circle, and shading the region to indicate which values of the variable are part of the solution set. By following these steps, we can solve a wide range of inequalities and represent their solution sets on a number line.

Key Takeaways

To solve an inequality, we need to follow the order of operations (PEMDAS) and isolate the variable.
We can represent the solution set on a number line by drawing a line and marking the point with a closed circle, and shading the region to indicate which values of the variable are part of the solution set.
Inequalities can be used to compare two or more values and are an essential part of algebra.

Practice Problems

Solve the inequality $2x - 5 \geq 3x + 2$ and represent the solution set on a number line.
Solve the inequality $x - 2 \leq 3x - 5$ and represent the solution set on a number line.
Solve the inequality $4x + 2 \geq 2x + 10$ and represent the solution set on a number line.

Answer Key

$x \leq 7$
$x \geq 3$
$x \geq 2$

In this article, we will answer some of the most frequently asked questions about solving inequalities and representing solution sets on a number line.

Q: What is an inequality?

A: An inequality is a statement that compares two or more values using a mathematical symbol such as <, >, ≤, or ≥.

Q: How do I solve an inequality?

A: To solve an inequality, you need to follow the order of operations (PEMDAS) and isolate the variable. This involves adding, subtracting, multiplying, or dividing both sides of the inequality by the same value to get the variable by itself.

Q: What is a solution set?

A: A solution set is the set of all values of the variable that satisfy the inequality.

Q: How do I represent a solution set on a number line?

A: To represent a solution set on a number line, you draw a line and mark the point with a closed circle if it is included in the solution set, or an open circle if it is not included. You then shade the region to the left or right of the point to indicate which values of the variable are part of the solution set.

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 indicates that the point is included in the solution set, while an open circle indicates that the point is not included.

Q: How do I determine whether to shade to the left or right of a point on a number line?

A: To determine whether to shade to the left or right of a point on a number line, you need to look at the inequality symbol. If the symbol is ≤ or ≥, you shade to the left of the point. If the symbol is < or >, you shade to the right of the point.

Q: Can I have multiple solution sets for a single inequality?

A: Yes, it is possible to have multiple solution sets for a single inequality. This occurs when the inequality has multiple solutions, such as when the inequality is true for both positive and negative values of the variable.

Q: How do I graph a solution set on a number line if the inequality has multiple solutions?

A: To graph a solution set on a number line if the inequality has multiple solutions, you need to draw multiple lines and mark the points with closed or open circles as needed. You then shade the regions to the left or right of the points to indicate which values of the variable are part of the solution set.

Q: Can I use a number line to solve inequalities with variables on both sides?

A: Yes, you can use a number line to solve inequalities with variables on both sides. However, you need to follow the steps outlined in the article and isolate the variable on one side of the inequality before representing the solution set on a number line.

Q: How do I know if an inequality is true or false?

A: To determine if an inequality is true or false, you need to substitute a value of the variable into the inequality and evaluate the expression. If the expression is true, the inequality is true. If the expression is false, the inequality is false.

Q: Can I use a number line to solve inequalities with absolute values?

A: Yes, you can use a number line to solve inequalities with absolute values. However, you need to follow the steps outlined in the article and isolate the variable on one side of the inequality before representing the solution set on a number line.

Conclusion

In this article, we answered some of the most frequently asked questions about solving inequalities and representing solution sets on a number line. We covered topics such as what an inequality is, how to solve an inequality, what a solution set is, and how to represent a solution set on a number line. We also discussed how to determine whether to shade to the left or right of a point on a number line, how to graph a solution set on a number line if the inequality has multiple solutions, and how to use a number line to solve inequalities with variables on both sides, absolute values, and multiple solutions. By following the steps outlined in this article, you can become proficient in solving inequalities and representing solution sets on a number line.