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# How to Graph Inequalities

An inequality in an equation is a mathematical statement that compares two values using an inequality symbol, such as < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to

• How to Graph Inequalities
• Solved Examples
• FAQs

An inequality in an equation is a mathematical statement that compares two values using an inequality symbol, such as < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). Inequalities express a relationship between two expressions, such as x > 5, meaning that x is greater than 5.

Like an equation, inequality can be solved by isolating the variable on one side of the inequality. However, the solution to an inequality is not a single value, but a range of values that make the inequality true.

Examples of inequalities include:

• x > 5 (x is greater than 5)
• y < 10 (y is less than 10)
• z ≥ 3 (z is greater than or equal to 3) You can also solve an inequality using the same operations as an equation, but you need to remember to flip the inequality symbol if you multiply or divide both sides by a negative number.

Example:

x + 3 > 7

x > 4 (subtract 3 from both sides of the inequality)

Another Example:

3x < 6

x < 2 (divide both sides by 3)

You can also graph the solution of an inequality on a number line, with a closed dot for “greater than or equal to” or “less than or equal to” and an open dot for “greater than” or “less than”

It is important to note that solving and graphing inequalities gives you the set of all the solutions that make the inequality true, rather than a single solution.

## How to Graph Inequalities

Graphing inequalities is a way to show the solutions to inequality on a number line. The solution for linear inequalities in two variables is an ordered pair that is true for the inequality statement. Let us say if Px + Qy > R is a linear inequality where x and y are two variables, then an ordered pair (x, y) satisfying the statement will be the required solution. Here’s how you can graph inequalities for an 8th grader:

1. Start by drawing a number line. This is the line where you will graph the solutions to the inequality.

2. We have to simplify the given linear inequality in the same way as we do in the case of linear equations. Also, for linear inequalities in two variables, we need two test points(for x and y).

3. Next, choose a test point. A test point is a point on the number line that you will use to test whether the inequality is true or false. You can choose any point on the number line as a test point, but it’s often easiest to choose 0 or 1.

4. Plug the test point into the inequality and solve. If the test point makes the inequality true, then the point is shaded in. If the test point makes the inequality false, then the point is not shaded in.

5. Use the inequality symbol to determine whether to shade above or below or left or right of the line. For example, if the inequality is “x > 3,” then you would shade in all the points to the right of 3 on the number line. If the inequality is “x < 4,” then you would shade in all the points to the left of 4 on the number line. For “greater than or equal to” or “less than or equal to” inequality use a closed dot or solid line and for “greater than” or “less than” use an open dot or shaded line.

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Here is an example of how to graph the inequality “x > 2”:

1. Draw a number line
2. Choose a test point, for example, x = 1
3. Plug the test point into the inequality, 1 > 2, which is false
4. Shade all the points to the right of 2, because x > 2 means “greater than”

Another example:

Graph the inequality “y ≤ -3”

1. Draw a number line
2. Choose a test point, for example, y = -4
3. Plug the test point into the inequality, -4 ≤ -3, which is true
4. Shade all the points the below -3, because y ≤ -3 means “less than or equal”

It’s important to remember that the solution of an inequality is the set of all the values that make the inequality true, not a single value like an equation.

## Order of Operations FAQS

##### How do I know which side of the inequality to shade on the graph?

When you graph an inequality, you need to use the inequality symbol to determine whether to shade above or below the line. For example, if the inequality is “x > 3,” then you would shade in all the points to the right of 3 on the number line. If the inequality is “x < 4,” then you would shade in all the points to the left of 4 on the number line.

##### How do I graph a compound inequality?

A compound inequality is an inequality that has more than one inequality symbol, such as “x > 2 and x < 5.” To graph a compound inequality, you need to graph each inequality separately and then find the intersection of the two solutions. The intersection is the set of all solutions that make both inequalities true.

##### How do I graph an inequality with a variable on both sides?

An inequality with a variable on both sides, such as “x + 2 > 3,” can be graphed in the same way as an inequality with a variable on one side. First, you need to get the variable on one side of the inequality by subtracting 2 from both sides. Then you graph it like any other inequality.

##### What does it mean if the inequality has an "or" sign?

An “or” sign in an inequality means that the solution set is the union of the solutions of the two inequalities. So for example, if you have an inequality like “x <2 or x > 3” it would mean that the solution set is all the solutions of x < 2 and all the solutions of x > 3 on the number line.

##### How do I graph an inequality with fractions?

An inequality with fractions can be graphed in the same way as an inequality without fractions. First, you need to clear the fractions by multiplying both sides of the inequality by the least common multiple of the denominators. Then you graph it like any other inequality.

##### How do I graph an inequality in two variables?

An inequality in two variables, such as “x + y > 3,” can be graphed on a coordinate plane. To graph the inequality, you can replace the inequality symbol with an equal sign and graph the equation as a straight line. Then you use the inequality symbol to determine which side of the line to shade. Gloria Mathew writes on math topics for K-12. A trained writer and communicator, she makes math accessible and understandable to students at all levels. Her ability to explain complex math concepts with easy to understand examples helps students master math. LinkedIn

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