Here you will learn about inequalities, including comparing quantities using inequalities, interpreting inequalities, representing inequalities, and solving inequalities.
Students first learn about inequality symbols in the first grade to represent relationships as part of their work with number and operations in base ten. They will expand upon this knowledge as they progress through elementary school into middle school, where they work with expressions and equations.
What are inequalities?
Inequalities are a comparison between two numbers, values, or expressions.
One of the quantities may be less than, greater than, less than or equal to, or greater than or equal to the other things.
This image describes the shape and direction of the symbols.
When stacking identical blocks, the height of 3 blocks is greater than the height of 1 block. The lines joining the stacks give the shape and direction of the inequality symbols.
The symbols used to make the comparisons are in the table below.
You can use inequalities to make comparisons between quantities.
You can represent inequalities on a number line.
Recall, that x = 5 is an equation. The variable x is only equal to 5. In an equality, the value of x can be many numbers.
You can interpret and represent inequality statements on the number line.
An open circle shows it does not include the value.
A closed circle shows it does include the value.
You can solve simple inequalities.
What are inequalities?
Common Core State Standards
How does this relate to 1st through 6th grade math?
Grade 1 – Number and Operations Base 10 (1.NBT.B.3) Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, \, =, and <.
Grade 2 – Number and Operations Base 10 (2.NBT.A.4) Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, \, =, and < symbols to record the results of comparisons.
Grade 3 – Number and Operations – Fractions (3.NF.A.3.d) Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, \, =, or <, and justify the conclusions, for example, by using a visual fraction model.
Grade 4 – Number and Operations Base Ten (4.NBT.A.2) Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, \, =, and < symbols to record the results of comparisons.
Grade 4 – Number and Operations – Fractions (4.NF.A.2) Compare two fractions with different numerators and different denominators, for example by creating common denominators or numerators, or by comparing to a benchmark fraction such as \frac{1}{2}. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, \, =, or <, and justify the conclusions, for example, by using a visual fraction model.
Grade 4 – Number and Operations – Fractions (4.NF.C.7) Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, \, =, or <, and justify the conclusions, for example, by using a visual model.
Grade 5 – Number and Operations Base Ten (5.NBT.A.3b) Compare two decimals to thousandths based on meanings of the digits in each place, using >, \, =, and < symbols to record the results of comparisons.
Grade 6 – Number System (6.NS.C.7.a) Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret –3 > –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right.
Grade 6 – Expressions and Equations (6.EE.B.8) Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.
Grade 6 – Expressions and Equations (6.EE.B.5) Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
Grade 7 – Expressions and Equations (7.EE.4b) Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid \$50 per week plus \$3 per sale. This week you want your pay to be at least \$100. Write an inequality for the number of sales you need to make, and describe the solutions.
How to compare quantities using inequalities
In order to compare quantities using inequalities:
Choose a form to represent the quantities.
Compare the quantities.
Write the comparison using the appropriate inequality symbol
[FREE] End of Year Math Assessments (Grade 4 & Grade 5)
Assess math progress for the end of grade 4 and grade 5 or prepare for state assessments with these mixed topic, multiple choice questions and extended response questions!
[FREE] End of Year Math Assessments (Grade 4 & Grade 5)
Assess math progress for the end of grade 4 and grade 5 or prepare for state assessments with these mixed topic, multiple choice questions and extended response questions!
The solution set is all the numbers less than or equal to -13.
Example 5: solve the inequality
Solve the inequality: -4x < -16
Choose one side of the inequality to have the variable alone.
Keep the variable on the left side of the inequality.
Use the additive inverse or multiplicative inverse to get the variable alone.
The multiplicative inverse of -4 is \cfrac{-1}{4}
\begin{aligned} & -4 x < -16 \\\\ & \cfrac{-4 x}{-4} \, < \, \cfrac{-16}{-4} \\\\ & x > 4 \end{aligned}
Change the direction of the inequality when dividing by -4.
Write your solution with the inequality symbol.
x > 4
Graph the solution set on the number line.
The solution set is all the numbers greater than 4.
Example 6: real world application of inequalities
Julie spends at least \$10.25 on each movie ticket. Write and graph an inequality representing this.
Identify the starting value.
The starting value on the number line is 10.25.
Decide if the starting value is included in the solution set.
10.25 will be in the solution set because the least amount of money Julie spends is \$10.25 , or she can spend more than that.
Identify the solution set with a straight line.
The inequality is: x ≥ 10.25
The solution set is all the numbers greater than or equal to 10.25.
For a detailed explanation, go to:
Step by step guide: Solving inequalities
Teaching tips for inequalities
To help students memorize the inequality symbols, make sure they are posted in the classroom or somewhere else easily accessible. Also encourage students to read inequalities both forward and backwards, which forces them to focus on the shape of the inequality symbol.
When introducing the ≤ and ≥, explicitly make the connection between these signs and the greater than, less than, and equal sign.
Number line worksheets are a good way to help students develop this skill, but be sure to also include activities that are more interactive. This could include graphing inequalities digitally or using things that happen in and around the classroom as a basis for writing and graphing real world inequalities.
Demonstrate to students using the number line why the inequality symbol flips when multiplying or dividing by a negative number.
Reinforce conceptual understanding using a number line between inequalities and equalities.
Easy mistakes to make
Confusing the inequality symbols For example, comparing 5 < 4 and reading it as “5 is greater than 4” .
Forgetting to flip the inequality symbol when dividing or multiplying by a negative number when solving the inequality For example,
The answer should be
Reading the inequality incorrectly from a number line For example, Reading this number line as x > 4 instead of x ≥ 4
Practice inequalities questions
1. Choose the correct symbol to make the comparison.
\le
<
>
\ge
Compare the fractions by finding the common denominator and use equivalent fractions to change the fractions.
Can you graph more than one inequality on a number line?
Yes, when you get into pre-algebra and algebra you will be graphing compound inequalities on the number line which is more than one inequality.
Is moving to the right always the greater than direction and moving left the less than direction?
Yes, looking at a number line, right is a positive or greater than direction and left is a negative or less than direction.
Do all inequalities only have one variable?
No, just like expressions and equations, inequalities increasingly get more complex as students advance in their math learning. In 7th grade, students will continue learning by working with linear inequalities, also known as two-step inequalities.
Why do you flip the inequality symbol when multiplying or dividing by a negative number?
The inequality symbol flips because you have to keep the inequality as a true comparison. For example, if you have 3 < 5 and you multiply both sides of the inequality by -1 you will get -3 < -5. This is no longer a true inequality comparison. In order to make it true, you will have to flip the inequality symbol, so < becomes >.
-3 < -5 becomes -3 > -5.
Is solving linear inequalities the same as solving linear equations?
Yes, the way you would solve a linear inequality follows the same process as you would do to solve a linear equation. With equations, you choose a side of the equation to bring the variable to, and you would do the same for an inequality. It’s most common to have the variable on the left hand side of an equality and the integers or real numbers on the right side.
Are there other ways to express inequalities or inequality signs?
Yes, when you get into high school math, you can use interval notation with parentheses to represent inequalities.
Are there only linear inequalities?
No, there are other types of inequalities, such as quadratic and polynomial inequalities and absolute value inequalities, that you will represent and solve in high school math.
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