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Place value Comparing fractions Decimal number lineOne step equations
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.
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.
\hspace{1.6cm} Comparison Symbols
Less than  
Greater than  
Less than or equal to  
Greater than or equal to  
Equal  
Not equal 
Step by step guide: Great than signs
Step by step guide: Less than signs
3 tens and 3 ones \hspace{2cm} 4 tens and 1 ones \bf{33} \hspace{1.6cm}\textbf{<} \hspace{1.6cm} \bf{41} \hspace{0.5cm} 
3 shaded parts out \hspace{2cm} 2 shaded parts out of 5 equal parts \hspace{2.5cm} of 5 equal parts \bf{\cfrac{3}{5}} \hspace{1.8cm}\textbf{>} \hspace{1.9cm} \bf{\cfrac{2}{5}} \hspace{0.3cm} 
40 \%<\cfrac{3}{5}

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.
\bf{\textbf{x} < 5}
The solution set is the numbers less  \bf{\textbf{x} \leq 5}
The solution set is the numbers less 
\bf{\textbf{x} > 5}
The solution set is all the numbers greater  \bf{\textbf{x} \geq 5}
The solution set is all the numbers greater 
\begin{aligned} x+4&>9 \\ \colorbox{#dcd0ff}{4} &\quad \colorbox{#dcd0ff}{4} \\ x+44&>94 \\ x&>5 \end{aligned} Represent the answer on the number line. The solution is: x is any number greater  \begin{aligned} x8&\geq 2 \\ \colorbox{#dcd0ff}{+8} &\quad \colorbox{#dcd0ff}{+8} \\ x8+8&\geq 2+8 \\ x&\geq 6 \end{aligned} Represent the answer on the number line. The solution is: x is any number greater 
\begin{aligned} 3x&<12 \\ \cfrac{3x}{3}&<\cfrac{12}{3} \\ x&<4 \end{aligned} Represent the answer on the number line. The solution is: x is any number lesser  \begin{aligned} \cfrac{x}{2} \, &\leq 2 \\ \colorbox{#dcd0ff}{× −2} &\quad \colorbox{#dcd0ff}{× −2} \\ x&\geq 4 \end{aligned} (flip the inequality symbol when multiplying The solution is: x is any number greater 
How does this relate to 1st through 6th grade math?
In order to compare quantities using inequalities:
Step by step guide: Converting fractions, decimals and percentages
In order to graph inequalities on a number line:
In order to solve one step inequalities:
Step by step guide: Solving inequalities
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!
DOWNLOAD FREEAssess 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!
DOWNLOAD FREECompare the fractions using a <, \, >, \, =.
Both numbers are in the same form, so leave them as fractions.
Get a common denominator.
The common denominator between \, \cfrac{2}{3} \, and \, \cfrac{1}{4} is 12.
\begin{aligned} &\cfrac{2}{3}=\cfrac{2 \, \times \, 4}{3 \, \times \, 4}=\cfrac{8}{12}\\\\ &\cfrac{1}{4}=\cfrac{1 \, \times \, 3}{4 \, \times \, 3}=\cfrac{3}{12} \end{aligned}
2Compare the quantities.
\cfrac{8}{12} \, is greater than \, \cfrac{3}{12}
3Write the comparison using the appropriate inequality symbol.
\cfrac{8}{12} \, > \, \cfrac{3}{12}
So, \cfrac{2}{3} \, > \, \cfrac{1}{4}
For a detailed explanation, go to:
Step by step guide: Comparing Fractions
Compare the fractions using a <, \, >, \, =.
Choose a form to represent the quantities.
Change the percent to a fraction.
34 \%=\cfrac{34}{100}
Make \, \cfrac{2}{5} \, have a denominator of 100.
\cfrac{2}{5}=\cfrac{2 \, \times \, 20}{5 \, \times \, 20}=\cfrac{40}{100}
Compare the quantities.
\cfrac{34}{100} \, is less than \, \cfrac{40}{100}
Write the comparison using the appropriate inequality symbol.
34 \% < \cfrac{2}{5}
For a detailed explanation, go to:
Step by step guide: Converting Fractions, Decimals, and Percents
Represent x \geq1
Identify the starting number.
The starting value on the number line is 1.
Place a closed circle or open circle on the number.
Draw a line from the starting number in the direction representing the inequality.
The graph starts at 1 with a closed circle and the arrow is pointing to the right showing all the numbers greater than 1 or equal to 1.
For a detailed explanation, go to:
Step by step guide: Inequalities on the number line
Stepbystep guide: Inequalities on a number line
Solve the inequality:
x+5 \leq8
Choose one side of the inequality to have the variable alone.
Keep the variable on the left side of the inequality symbol.
Use the additive inverse or multiplicative inverse to get the variable alone.
The additive inverse of +5 is 5 because + 55 = 0
\begin{aligned}
& x+5 \leq8 \\\\
& x+55 \leq85 \\\\
& x \leq13
\end{aligned}
Write your solution with the inequality symbol.
x \leq13
Graph the solution set on the number line.
The solution set is all the numbers less than or equal to 13.
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.
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
1. Choose the correct symbol to make the comparison.
Compare the fractions by finding the common denominator and use equivalent fractions to change the fractions.
The common denominator is 14.
\begin{aligned} & \cfrac{2}{7}=\cfrac{2 \, \times \, 2}{7 \, \times \, 2}=\cfrac{4}{14} \\\\ & \cfrac{1}{2}=\cfrac{1 \, \times \, 7}{2 \, \times \, 7}=\cfrac{7}{14} \\\\ & \cfrac{4}{14} \, < \, \cfrac{7}{14} \end{aligned}
2. Choose the graph that represents the inequality.
x \ge1
x \ge1 is read as ‘x is greater than or equal to 1’
1 is in the solution set so the circle is closed. The arrow points towards the right because that represents the numbers greater than.
3. Solve the inequality:
x7 < 4
Keep the variable on the left side of the inequality symbol. Use the additive inverse to solve the inequality.
The additive inverse of 7 is + 7 because 7 + 7 = 0. In other words, add 7 to both sides of the inequality.
\begin{aligned} & x7<4 \\\\ & x7+7<4+7 \\\\ & x<11 \end{aligned}
4. Solve the inequality:
5 x \geq 20
Keep the variable on the left side of the inequality symbol. Use the multiplicative inverse to solve the inequality.
The multiplicative inverse is 5 \times \cfrac{1}{5}=1. In other words, divide each side of the inequality by 5.
\begin{aligned} & 5 x>20 \\\\ & \cfrac{5 x}{5} > \cfrac{20}{5} \\\\ & x>4 \end{aligned}
5. Which number line represents the solution set to the following inequality?
x + 3 < 4
Keep the variable on the left side of the inequality symbol. Use the additive inverse to solve the inequality.
The additive inverse of +3 is 3 because 33 = 0. In other words, subtract 3 from both sides of the inequality.
\begin{aligned} & x+3<4 \\\\ & x+33<43 \\\\ & x<1 \end{aligned}
6. Two times a number is greater than 6. Find the solution set.
Translate the problem into an inequality.
2 x>6
Solve the inequality. Keep the variable on the left side of the inequality symbol.
Use the multiplicative inverse to solve.
\begin{aligned} & 2 x>6 \\\\ & \cfrac{2 x}{2}>\cfrac{6}{2} \\\\ & x>3 \end{aligned}
Yes, when you get into prealgebra and algebra you will be graphing compound inequalities on the number line which is more than one inequality.
Yes, looking at a number line, right is a positive or greater than direction and left is a negative or less than direction.
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 twostep inequalities.
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.
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.
Yes, when you get into high school math, you can use interval notation with parentheses to represent 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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