Match Each Inequality To The Number Line That Represents Its Solution.1. $x - 99 \leq -104$2. $x - 51 \leq -43$3. $150 + X \leq 144$4. $75 \ \textless \ 69 - X$
Introduction
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. One way to visualize and solve inequalities is by using a number line. A number line is a line that represents all the real numbers, with each point on the line corresponding to a specific value. In this article, we will explore how to match each inequality to the number line that represents its solution.
Understanding Inequalities
Before we can solve inequalities on a number line, we need to understand what an inequality is. An inequality is a statement that compares two or more values using a mathematical symbol, such as <, >, ≤, or ≥. For example, the inequality x - 99 ≤ -104 is a statement that compares the value of x to -104.
Solving Inequalities on a Number Line
To solve an inequality on a number line, we need to follow these steps:
- Write the inequality: Write the inequality in the form ax + b ≤ c, where a, b, and c are constants.
- Solve for x: Solve the inequality for x by isolating x on one side of the inequality.
- Graph the solution: Graph the solution on a number line by marking the point that represents the solution.
Example 1: x - 99 ≤ -104
Let's solve the inequality x - 99 ≤ -104 on a number line.
Step 1: Write the inequality
The inequality is already written in the form ax + b ≤ c: x - 99 ≤ -104.
Step 2: Solve for x
To solve for x, we need to isolate x on one side of the inequality. We can do this by adding 99 to both sides of the inequality:
x - 99 + 99 ≤ -104 + 99
This simplifies to:
x ≤ -5
Step 3: Graph the solution
To graph the solution on a number line, we need to mark the point that represents the solution. Since x ≤ -5, we will mark the point -5 on the number line.
Example 2: x - 51 ≤ -43
Let's solve the inequality x - 51 ≤ -43 on a number line.
Step 1: Write the inequality
The inequality is already written in the form ax + b ≤ c: x - 51 ≤ -43.
Step 2: Solve for x
To solve for x, we need to isolate x on one side of the inequality. We can do this by adding 51 to both sides of the inequality:
x - 51 + 51 ≤ -43 + 51
This simplifies to:
x ≤ 8
Step 3: Graph the solution
To graph the solution on a number line, we need to mark the point that represents the solution. Since x ≤ 8, we will mark the point 8 on the number line.
Example 3: 150 + x ≤ 144
Let's solve the inequality 150 + x ≤ 144 on a number line.
Step 1: Write the inequality
The inequality is already written in the form ax + b ≤ c: 150 + x ≤ 144.
Step 2: Solve for x
To solve for x, we need to isolate x on one side of the inequality. We can do this by subtracting 150 from both sides of the inequality:
150 + x - 150 ≤ 144 - 150
This simplifies to:
x ≤ -6
Step 3: Graph the solution
To graph the solution on a number line, we need to mark the point that represents the solution. Since x ≤ -6, we will mark the point -6 on the number line.
Example 4: 75 < 69 - x
Let's solve the inequality 75 < 69 - x on a number line.
Step 1: Write the inequality
The inequality is already written in the form ax + b ≤ c: 75 < 69 - x.
Step 2: Solve for x
To solve for x, we need to isolate x on one side of the inequality. We can do this by subtracting 69 from both sides of the inequality:
75 - 69 < 69 - x - 69
This simplifies to:
6 < -x
Step 3: Graph the solution
To graph the solution on a number line, we need to mark the point that represents the solution. Since 6 < -x, we will mark the point -7 on the number line.
Conclusion
In this article, we have explored how to match each inequality to the number line that represents its solution. We have used four examples to demonstrate how to solve inequalities on a number line. By following the steps outlined in this article, you can solve any inequality on a number line.
Key Takeaways
- Inequalities are used to compare two or more values.
- A number line is a line that represents all the real numbers.
- To solve an inequality on a number line, we need to write the inequality, solve for x, and graph the solution.
- The solution to an inequality is the set of all values that satisfy the inequality.
Practice Problems
- Solve the inequality x + 2 ≤ 5 on a number line.
- Solve the inequality 3x - 2 ≤ 7 on a number line.
- Solve the inequality x - 4 ≥ 2 on a number line.
- Solve the inequality 2x + 1 ≥ 3 on a number line.
Answer Key
- x ≤ 3
- x ≤ 3
- x ≥ 6
- x ≥ 1
Frequently Asked Questions (FAQs) about Solving Inequalities on a Number Line ====================================================================================
Q: What is a number line?
A: A number line is a line that represents all the real numbers. It is a visual representation of the number system, with each point on the line corresponding to a specific value.
Q: How do I solve an inequality on a number line?
A: To solve an inequality on a number line, you need to follow these steps:
- Write the inequality: Write the inequality in the form ax + b ≤ c, where a, b, and c are constants.
- Solve for x: Solve the inequality for x by isolating x on one side of the inequality.
- Graph the solution: Graph the solution on a number line by marking the point that represents the solution.
Q: What is the difference between a number line and a graph?
A: A number line is a line that represents all the real numbers, while a graph is a visual representation of a function or a set of points. A number line is used to solve inequalities, while a graph is used to visualize the behavior of a function.
Q: Can I use a number line to solve equations?
A: No, a number line is used to solve inequalities, not equations. Equations are solved using algebraic methods, such as adding, subtracting, multiplying, and dividing both sides of the equation.
Q: How do I graph the solution to an inequality on a number line?
A: To graph the solution to an inequality on a number line, you need to mark the point that represents the solution. If the inequality is of the form x ≤ a, you will mark the point a on the number line. If the inequality is of the form x ≥ a, you will mark the point a on the number line and shade the region to the right of the point.
Q: Can I use a number line to solve systems of inequalities?
A: Yes, you can use a number line to solve systems of inequalities. A system of inequalities is a set of two or more inequalities that must be satisfied simultaneously. To solve a system of inequalities using a number line, you need to graph the solution to each inequality on the number line and find the region where all the inequalities are satisfied.
Q: How do I find the solution to a system of inequalities on a number line?
A: To find the solution to a system of inequalities on a number line, you need to follow these steps:
- Graph the solution to each inequality: Graph the solution to each inequality on the number line.
- Find the intersection of the solutions: Find the region where all the inequalities are satisfied.
- Mark the solution: Mark the solution on the number line.
Q: Can I use a number line to solve linear programming problems?
A: Yes, you can use a number line to solve linear programming problems. Linear programming is a method of solving optimization problems that involve linear inequalities. To solve a linear programming problem using a number line, you need to graph the solution to each inequality on the number line and find the region where all the inequalities are satisfied.
Q: How do I find the solution to a linear programming problem on a number line?
A: To find the solution to a linear programming problem on a number line, you need to follow these steps:
- Graph the solution to each inequality: Graph the solution to each inequality on the number line.
- Find the intersection of the solutions: Find the region where all the inequalities are satisfied.
- Mark the solution: Mark the solution on the number line.
Conclusion
In this article, we have answered some frequently asked questions about solving inequalities on a number line. We have covered topics such as the definition of a number line, how to solve inequalities on a number line, and how to graph the solution to an inequality on a number line. We have also discussed how to use a number line to solve systems of inequalities and linear programming problems.