# How to graph inequalities on a number line

- Author: Noreen Niazi
- Last Updated on: August 22, 2023

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ToggleAs a math student, one of the essential skills you need to learn is how to graph inequalities on a number line. You will encounter this fundamental concept in algebra, calculus, and other advanced math courses. Inequalities are mathematical expressions that compare two quantities using the symbols <, >, ≤, or ≥.

Graphing inequalities on a number line visually represents the solutions to these expressions. In this tutorial, I will teach you how to graph inequalities on a number line like a pro.

## What are inequalities, and why do we graph them on a number line?

Mathematical expressions known as inequality provide a range of possible values for a variable. For instance, the inequality x > 2 indicates that any value larger than 2 can be assigned to the variable x. Inequalities are employed in a variety of real-world situations, such as figuring out the minimum and maximum temperatures in a specific location or the range of potential stock price values.

Visualising the range of values that a variable can have is possible by graphing inequalities on a number line. It is a helpful tool for deciphering inequality’s answer and resolving issues regarding disparities. You may quickly identify the values of the variable that satisfy an inequality by graphing it on a number line.

## Understanding the basics of number lines and their properties

A number line is a simple straight line that depicts every possible value of a variable. Each segment on the number line represents a range of values. The segments are denoted by numbers or tick marks, which represent the variable’s value on the number line.

When graphing inequalities, there are a few crucial features of the number line to comprehend. The first characteristic is that the numbers on the number line rise from left to right. The second characteristic is that the same amount of space separates any two numbers on the number line. The third characteristic is that the number line is limitless, continuing indefinitely in both directions.

Read more about other math topics

Mastering Percent Word Problems In 2023: Tips And Tricks For Success – LearnAboutMath

## Steps to graphing inequalities on a number line

To graph an inequality on a number line, follow these steps:

- Identify the variable in the inequality.
- Determine the range of values that the variable can take.
- Draw a number line and label the endpoints with the minimum and maximum values of the variable.
- Mark the endpoint(s) with a closed or open dot, depending on whether the endpoint is included in the range of values.
- Shade the region of the number line representing the range of values satisfying the inequality.

Let’s consider an example to illustrate these steps. Suppose we want to graph the inequality x > 2. The variable in this inequality is x, and the range of values that x can take is greater than 2. We draw a number line to graph this inequality on a number line and label the endpoint at 2 with an open dot. We then shade the region of the number line to the right of 2, since all values greater than 2 satisfy the inequality.

## Examples of graphing linear inequalities on a number line

Linear inequalities are inequalities that involve linear expressions. They are represented by straight lines on a number line. To graph a linear inequality on a number line, follow the steps outlined in the previous section.

Let’s consider an example of graphing a linear inequality. Suppose we want to graph the inequality y ≤ 2x + 1. We can rewrite this inequality as y – 2x ≤ 1.

To graph this inequality on a number line, we first graph the line y = 2x + 1. We then shade the region below the line since all values of (x,y) that satisfy the inequality lie below the line.

## Graphing quadratic inequalities on a number line

Quadratic inequalities are inequalities that involve quadratic expressions. Curves on a number line represent them. To graph a quadratic inequality on a number line, follow the steps outlined in the previous section.

Let’s consider an example of graphing a quadratic inequality. Suppose we want to graph the inequality y > x^2 – 4. We first graph the y = x^2 – 4 curve to graph this inequality on a number line. We then shade the region above the curve since all values of (x,y) that satisfy the inequality lie above the curve.

## Solving quadratic inequalities and graphing them on a number line

To solve a quadratic inequality, we first find the zeros of the quadratic expression. We then use these zeros to divide the number line into intervals. We test the inequality in each interval to determine the sign of the expression in that interval. We then shade the intervals that satisfy the inequality.

Let’s consider an example of solving and graphing a quadratic inequality. Suppose we want to solve the inequality x^2 – 4x > 0. We can factor this expression as x(x – 4) > 0. The zeros of this expression are x = 0 and x = 4.

We divide the number line into three intervals: (-∞, 0), (0, 4), and (4, ∞). We test the inequality in each interval and find that it is satisfied in the intervals (-∞, 0) and (4, ∞). We shade these intervals on the number line to get the inequality graph.

## How to use a graphing calculator to graph inequalities on a number line

Graphing calculators are powerful tools that can be used to graph inequalities on a number line. Most graphing calculators have a built-in function for graphing inequalities. To use a graphing calculator to graph an inequality, follow these steps:

- Press the “Y=” button on the calculator.
- Enter the inequality into the first function slot.
- Press “GRAPH” to graph the inequality.

## Tips and tricks for mastering graphing inequalities on a number line

It can be difficult to graph inequalities on a number line, especially when working with more intricate expressions. The following hints and techniques will help you become an expert at graphing inequalities on a number line:

**Develop your skills through practice.**You will get more adept at graphing inequalities on a number line as you practice.**Divide complicated expressions into simpler components**. This can assist you in determining the variable’s possible range of values and make it simpler to graph the inequality.**Draw straight lines and tick marks using a ruler.**You might make a clean and precise number line using this.**Verify your work.**Verify again that you have shaded the proper area of the number line on your graph.

## Conclusion: Why graphing inequalities on a number line is essential for math students.

Math students need to be able to graph inequality on a number line. You will come across it as a fundamental idea in algebra, calculus, and other higher maths courses. You will be able to resolve issues with disparities and comprehend these issues’ solutions more clearly if you become proficient in this skill. To master graphing inequalities on a number line, keep in mind that practise, breaking down complicated formulas, using a ruler, and double-checking your work are all essential.

Put your abilities to the test now that you know how to graph inequalities on a number line like a pro. To ensure you get the ideas, practice graphing various inequalities and double-check your work. Please don’t hesitate to contact us if you require any additional help or have any questions.

## FAQs: How to graph inequality on number line.

To graph inequalities with two numbers on a number line, you must:

- If necessary, find the x value for each inequality.
- Locate the values of x on the number line that make each inequality true.

Indicate each value with a circle. - Complete the circle if the inequality contains the phrase equal to (or). Leave the circle empty if not (> or ).
- Each circle should be used to create an arrow pointing toward the values that support the inequality.
- The arrow points to the left if either or is the inequality.
- The arrow points right when the inequality is > or.
- The location where both inequalities are true can be found by finding the intersection of the two arrows.
- This is the answer to the inequality system.

x

<–|—|—|—|—|—|—|—|—|—|–>

-5 -4 -3 -2 -1 0 1 2 3 4

●══════○

To graph $$x< 3$$ and $$x >=-2$$, for instance, follow these steps:

- No need to reorder the inequalities because they have already been solved for x.
- Draw circles over the numbers 3 and -2 on the number line.
- Leave the circle at three unfilled because x< 3 does not include equal to. Fill in the circle at -2 since x -2 contains equal to.
- As x< 3 denotes that x is smaller than 3, an arrow should be drawn from 3 to the left. Since x -2 indicates that x is greater than or equal to -2, an arrow should be drawn from -2 to the right.
- Find the point between -2 and 3 (not including 3) where the arrows overlap. This is the answer to the system of inequalities.

Using the procedures below, you can graph an inequality:

- When graphing the equation, consider the signs, >,, or as a = sign.
- Graph the equation as a dotted line if the inequality is <either or>.
- Graph the inequality as a solid line if the inequality is either $$<=or>=$$.
- Then, enter the (x, y) values into the inequality as at least one ordered pair on each side of the boundary line.
- Use a solid line to represent the boundary line if any points on it are solutions.
- Use a dotted line for the boundary line if any of the points on it aren’t solutions.
- From the circle, a line should be drawn toward the necessary numbers to make the inequality true.

For instance, you can do the following to graph y <2x + 3:

- Since the inequality is<, draw y = 2x + 3 as a dotted line on the graph.
- In the inequality, enter a point at either end of the line, such as (0, 0) or (2, 7).
- The shade on the side of the line if the point proves the disparity.
- If not, cover the opposite side.

We obtain (0, 0), which is true, 0 2(0) + 3. - Therefore, we shade the (0, 0) side.

We have 7 2(2) + 3, which is incorrect for (2, 7). We, therefore, do not shade that side.