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Here you will learn about solving inequalities, including how to solve linear inequalities, identify integers in the solution set, and represent solutions on a number line.
Students will first learn about solving simple inequalities as part of expressions and equations in 6th grade math and expand that knowledge in 7th grade math.
Solving inequalities allows you to calculate the values of an unknown variable in an inequality.
Solving inequalities is similar to solving equations, but where an equation has one unique solution, an inequality has a range of solutions.
In order to solve an inequality, you need to balance the inequality on each side of the inequality sign in the same way as you would balance an equation on each side of the equal sign. Solutions can be integers, decimals, positive numbers, or negative numbers.
For example,
Solving an equation  Solving an inequality 

Solve \begin{aligned} 2x + 1 &= 9 \\ 2x &= 8 \\ x&= 4 \end{aligned} 4 is the only solution to this  Solve \begin{aligned} 2x + 1 &< 9 \\ 2x &< 8 \\ x&< 4 \end{aligned} x can be any value that is less

Multiplying and dividing by a negative number 

This changes the direction of the inequality sign.
x can be any value that is greater than 4. 
How does this relate to 6th grade and 7th grade math?
In order to solve 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 FREESolve:
4 x+6<26
In this case you are subtracting 6 from both sides.
\begin{aligned} 4x+6&<26\\ 4x&<20 \end{aligned}
This leaves 4x on the left side of the inequality sign and 20 on the right side.
2Rearrange the inequality by dividing by the \textbf{x} coefficient so that \textbf{‘x’} is isolated.
In this case you need to divide both sides by 4.
\begin{aligned} 4x&<20\\ x&<5 \end{aligned}
This leaves x on the left side of the inequality sign and 5 on the right side.
3Write your solution with the inequality symbol.
x < 5
Any value less than 5 satisfies the inequality.
Solve:
5x4 \geq 26
Rearrange the inequality so that all the unknowns are on one side of the inequality sign.
In this case you need to add 4 to both sides.
\begin{aligned} 5x4&\geq26\\ 5x&\geq30 \end{aligned}
This leaves 5x on the left side of the inequality sign and 30 on the right side.
Rearrange the inequality by dividing by the \textbf{x} coefficient so that \textbf{‘x’} is isolated.
In this case you need to divide both sides by 5.
\begin{aligned} 5x&\geq30\\ x&\geq6 \end{aligned}
This leaves x on the left side of the inequality sign and 6 on the right side.
Write your solution with the inequality symbol.
x\geq6
Any value greater than or equal to 6 satisfies the inequality.
Solve:
3(x4)\leq12
Rearrange the inequality so that all the unknowns are on one side of the inequality sign.
Let’s start by expanding the parentheses.
3x12\leq12
Then you need to add 12 to both sides.
\begin{aligned}
3x12&\leq12\\
3x&\leq24
\end{aligned}
This leaves 3x on the left side of the inequality sign and 24 on the right side.
Rearrange the inequality by dividing by the \textbf{x} coefficient so that \textbf{‘x’} is isolated.
In this case you need to divide both sides by 3.
\begin{aligned} 3x&\leq24\\ x&\leq8 \end{aligned}
This leaves x on the left side of the inequality sign and 8 on the right side.
Write your solution with the inequality symbol.
x\leq8
Any value less than or equal to 8 satisfies the inequality.
Solve:
5x6 > 2x + 15
Rearrange the inequality so that all the unknowns are on one side of the inequality sign.
In this case you need to subtract 2x from both sides.
\begin{aligned}
5x6&>2x+15\\
3x6&>15
\end{aligned}
This leaves 3x6 on the left side of the inequality sign and 15 on the right side.
Rearrange the inequality so that \textbf{‘x’} s are on one side of the inequality sign and numbers on the other.
In this case you need to add 6 to both sides.
\begin{aligned}
3x6&>15\\
3x&>21
\end{aligned}
This leaves 3x on the left side of the inequality sign and 21 on the right side.
Rearrange the inequality by dividing by the \textbf{x} coefficient so that \textbf{‘x’} is isolated.
In this case you need to divide both sides by 3.
\begin{aligned}
3x&>21\\
x&>7
\end{aligned}
This leaves x on the left side of the inequality sign and 7 on the right side.
Write your solution with the inequality symbol.
x > 7
Any value greater than 7 satisfies the inequality.
Solve:
\cfrac{x+3}{5}<2
Rearrange the inequality to eliminate the denominator.
In this case you need to multiply both sides by 5.
\begin{aligned}
\cfrac{x+3}{5}&<2\\
x+3&<10
\end{aligned}
Rearrange the inequality so that \textbf{‘x’} s are on one side of the inequality sign and numbers on the other.
In this case you need to subtract 3 from both sides.
\begin{aligned}
\cfrac{x+3}{5}&<2\\
x+3&<10\\
x&<7
\end{aligned}
Write your solution with the inequality symbol.
x < 7
Any value less than 7 satisfies the inequality.
Solve:
6x+1\geq4
Rearrange the inequality so that all the unknowns are on one side of the inequality sign.
In this case you need to subtract 6 from both sides.
\begin{aligned}
6x+1&\geq4\\
6x&\geq3
\end{aligned}
Rearrange the inequality by dividing by the \textbf{x} coefficient so that \textbf{‘x’} is isolated.
In this case you need to divide both sides by 6.
\begin{aligned}
6x+1&\geq4\\
6x&\geq3\\
x&\geq\cfrac{3}{6}
\end{aligned}
This can be simplified to \, \cfrac{1}{2} \, or the decimal equivalent.
Write your solution with the inequality symbol.
x\geq\cfrac{1}{2}
Any value greater than or equal to \, \cfrac{1}{2} \, satisfies the inequality.
Represent the solution on a number line:
2x7 < 5
Rearrange the inequality so that \textbf{‘x’} s are on one side of the inequality sign and numbers on the other.
In this case you need to add 7 to both sides.
\begin{aligned} 2x7&<5\\ 2x&<12 \end{aligned}
Rearrange the inequality by dividing by the \textbf{x} coefficient so that \textbf{‘x’} is isolated.
In this case you need to divide both sides by 2.
\begin{aligned}
2x7& <5\\
2x& <12\\
x& < 6
\end{aligned}
Represent your solution on a number line.
x <6
Any value less than 6 satisfies the inequality. An open circle is required at 6 and the values lower than 6 indicated with an arrow.
Solve:
12x <7
Rearrange the inequality so that \textbf{‘x’} s are on one side of the inequality sign and numbers on the other.
In this case you need to subtract 1 from both sides.
\begin{aligned} 12x & <7 \\ 2x & <6 \end{aligned}
Rearrange the inequality by dividing by the \textbf{x} coefficient so that \textbf{‘x’} is isolated.
In this case you need to divide both sides by negative 2.
6 \div2=3
Change the direction of the inequality sign.
Because you divided by a negative number, you also need to change the direction of the inequality sign.
\begin{aligned} 12x & <7 \\ 2x & <6 \\ x &>3 \end{aligned}
List the integer values that satisfy:
3 < x+1\leq8
Rearrange the inequality so that all the unknowns are on one side of the inequality sign.
In this case you need to subtract 1 from each part.
\begin{aligned} 3&<x+1\leq8\\ 2&<x\leq7\\ \end{aligned}
List the integer values satisfied by the inequality.
2<x\leq7
2 is not included in the solution set. 7 is included in the solution set. The integers that satisfy this inequality are:
3, 4, 5, 6, 7
List the integer values that satisfy:
7\leq4x\leq20
Rearrange the inequality so that all the unknowns are on one side of the inequality sign.
In this case you need to divide each part by 4.
\begin{aligned} 7\leq \, & 4x\leq20\\ \cfrac{7}{4}\leq & \; x \leq5 \end{aligned}
List the integer values satisfied by the inequality.
\cfrac{7}{4} \leq x \leq 5
\cfrac{7}{4} \, is included in the solution set but it is not an integer.
The first integer higher is 2.
5 is also included in the solution set.
The integers that satisfy this inequality are:
2, 3, 4, 5
List the integer values that satisfy:
3<2x+5\leq7
Rearrange the inequality so that all the unknowns are on one side of the inequality sign.
In this case you need to subtract 5 from each part.
\begin{aligned}
3<2x+5&\leq7\\
8<2x&\leq2
\end{aligned}
Rearrange the inequality so that \textbf{‘x’} is isolated. In this case you need to divide each part by \bf{2} .
\begin{aligned} 3<2x+5&\leq7\\ 8<2x&\leq2\\ 4<x&\leq1 \end{aligned}
Represent the solution set on a the number line
4<x\leq1
4 is not included in the solution set so requires an open circle. 1 is included in the solution set so requires a closed circle.
Put a solid line between the circles to indicate all the values that satisfy the solution set.
1. Solve:
3x+7 < 31
\begin{aligned} 3x+7&<31\\ 3x&<24\\ x&<8 \end{aligned}
2. Solve:
4x3\geq25
\begin{aligned} 4x3&\geq25\\ 4x&\geq28\\ x&\geq7 \end{aligned}
3. Solve:
2(x5)\leq8
\begin{aligned} 2(x5)&\leq8\\ 2x10&\leq8\\ 2x&\leq18\\ x&\leq9 \end{aligned}
4. Solve:
6x5 > 4x + 1
\begin{aligned} 6x5&>4x+1\\ 2x5&>1\\ 2x&>6\\ x&>3 \end{aligned}
5. Solve:
\cfrac{x4}{2}>6
\begin{aligned} \cfrac{x4}{2}&>6\\ x4&>12\\ x&>16 \end{aligned}
6. Solve:
8x+1\geq3
\begin{aligned} 8x+1&\geq3\\ 8x&\geq2\\ x&\geq\cfrac{2}{8}\\ x&\geq\cfrac{1}{4} \end{aligned}
7. Represent the solution on a number line.
5x2 < 28
\begin{aligned} 5x2&<28\\ 5x&<30\\ x&<6 \end{aligned}
An open circle is required and all values less than 6 indicated.
8. Solve:
23x > 14
\begin{aligned} 23x &>14 \\ 3x &>12 \\ x &< 4 \end{aligned}
Change the direction of the inequality sign as you have divided by a negative number.
9. List the integer values that satisfy:
2<x+3\leq5
\begin{aligned} 2<x&+3\leq5\\ 1< \, &x\leq2 \end{aligned}
1 is not included in the solution set as is greater than 1.
2 is included in the solution set as x is less than or equal to 2.
10. List the integer values that satisfy:
4\leq3x\leq21
\begin{aligned} 4\leq3&x\leq21\\ \cfrac{4}{3} \, \leq \, &x\leq7 \end{aligned}
The first integer greater than \, \cfrac{4}{3} \, is 2.
7 is included in the solution set as x is less than or equal to 7.
11. List the integer values that satisfy:
4<3x+2\leq5
\begin{aligned} 4<3x&+2\leq5\\ 6<3&x\leq3\\ 2< \, &x\leq1 \end{aligned}
2 is not included in the solution set as x is greater than 2.
1 is included in the solution set as x is less than or equal to 1.
Solving inequalities is where you calculate the values that an unknown variable can be in an inequality.
To solve an inequality, you need to balance the inequality on each side of the inequality sign in the same way as you would balance an equation on each side of the equal sign. Solutions can be integers, decimals, positive numbers, or negative numbers.
Solving inequalities is similar to solving equations, but where an equation has one unique solution, an inequality has a range of solutions.
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Prepare for math tests in your state with these Grade 3 to Grade 6 practice assessments for Common Core and state equivalents.
40 multiple choice questions and detailed answers to support test prep, created by US math experts covering a range of topics!