Solving Linear Inequalities Worksheets with Solutions

Solving linear inequalities is a foundational skill that you will build throughout your studies of elementary and high school algebra. While it isn’t the hardest concept you will encounter in the entire class, there are a few tips and tricks you should get used to. And the best way to do that is by practicing! 

That’s why I put together this collection of solving linear inequalities worksheets! Let’s dig in so that you can master this important algebra concept!

What is a Linear Inequality?

A linear inequality is a type of statement in algebra that compares two linear equations. To make this comparison, we place an inequality symbol in between the two sides of the inequality. For example, \(x+1>2x-1\).

Take a moment to review this inequality symbol list:

  • < (less than)
  • > (greater than)
  • ≤ (less than or equal to)
  • ≥ (greater than or equal to)

Linear inequalities are similar to linear equations, however instead of an equals sign, a linear inequality contains one of the inequality symbols from the above list.

How to Solve a Linear Inequality

The steps that you take to solve a linear inequality are a very similar to those that you would take while solving an algebraic equation. When solving linear inequalities, you often perform similar operations as you would when solving linear equations

Just like in linear equations, you can add or subtract terms on both sides, and multiply or divide by constants on both sides. However, unlike solving linear equations, linear inequalities have one very important rule:

Remember: If you multiply or divide both sides of an inequality by a negative number, the inequality sign must be flipped! 

You will see some examples of this in the solving linear inequalities worksheets linked below!

Another important difference between solving a linear equation and solving a linear inequality is that the solution to a linear inequality will be the set of all values that make the inequality true (rather than just a single value that makes the equation true).

The solution to a linear inequality will tell you the places on the graph where one line is greater than or less than the other.

For example, consider the coordinate plane below that shows two lines in slope-intercept form. 

the graph of x+1 shown in red, the graph of 2x-1 shown in green

As you can see, the red line is above the green line as long as \(x<2\). After this, the green line is above the red line. So if we wanted to write the solution to \(x+1>2x-1\), we would say that the solution set is \(x<2\), or all x-values less than 2 (note that we do not include 2 in the solution set since that is the point where the lines are equal).

Solving Algebraically

In order to solve \(x+1>2x-1\) algebraically, we will use the same algebra strategies that we use when solving linear equations. We will collect all x-terms on the left side of the equation, and all non-variable terms on the right side of the equation.

\begin{split} x+1&>2x-1 \\ \\ -x&>-2 \\ \\ x&<2 \end{split}

Since we had to divide both sides by -1, notice that we flipped the sign of the inequality symbol! Therefore, the solution to this inequality is x<2 (which we can confirm using the coordinate plane above).

Another way that we can represent this solution is on a number line. We draw a line to the left of 2 with a hollow circle at 2 to indicate that 2 is not part of our solution set. We would use a filled in dot to indicate a value that is part of the solution set.

the solution to two different linear inequalities represented on a number line

There are many ways to solve linear inequalities and represent their solutions. In the math worksheets that follow below, you will focus mainly on solving algebraically and representing your answer on a number line!

One-Step Inequalities Math Worksheet

To get started, try the set of one-step inequalities in the worksheet linked below. These problems will help you become familiar with solving a linear equality algebraically, and representing the solution on a number line. 

I have created this worksheet in PDF format for your convenience. Remember to check the answer key provided to make sure that you understand the basics of solving linear inequalities.

Download the PDF worksheet by clicking below!

Multi-Step Inequalities Math Worksheet

Now that you have had some practice with solving one-step inequality problems, use the following worksheet to try your hand at some more complex linear inequality problems. This worksheet contains two-step inequalities as well as some more difficult multi-step linear inequality problems! 

This worksheet is also provided in PDF format for ease of access. Remember to check the answer key to confirm your understanding of solving more complex linear inequalities.

Download the PDF worksheet by clicking below!

Practice Solving Linear Inequalities

Inequalities may seem scary at first because of the inequality symbol, but as you have seen, they are actually solved in a very similar way to linear equations.

Use the math worksheets linked above to practice getting comfortable with solving a variety of linear inequality problems. Like any math concept, the more you practice the more familiar the concept will start to feel.

I hope these solving linear inequalities worksheets have helped you practice your inequality solving skills, as well as representing the solution set on a number line!

Ready to start applying your skills? Check out this linear inequalities word problems worksheet!

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