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Elementary math Algebra Inequalities

Solving inequalities

Solving inequalities

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.

What is solving inequalities?

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 Inequalities image 1

What is solving inequalities?

What is solving inequalities?

Common Core State Standards

How does this relate to 6th grade and 7th grade math?

  • 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 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 solve inequalities

In order to solve inequalities:

  1. Rearrange the inequality so that all the unknowns are on one side of the inequality sign.
  2. Rearrange the inequality by dividing by the \textbf{x} coefficient so that \textbf{‘x’} is isolated.
  3. Write your solution with the inequality symbol.

[FREE] End of Year Math Assessments (Grade 4 & Grade 5)

[FREE] End of Year Math Assessments (Grade 4 & Grade 5)

[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!

DOWNLOAD FREE
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[FREE] End of Year Math Assessments (Grade 4 & Grade 5)

[FREE] End of Year Math Assessments (Grade 4 & Grade 5)

[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!

DOWNLOAD FREE

Solving linear inequalities examples

Example 1: solving linear inequalities

Solve:

Solving Inequalities example 1 image 1

  1. Rearrange the inequality so that all the unknowns are on one side of the inequality sign.

In this case you are subtracting 6 from both sides.

Solving Inequalities example 1 image 2

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.

Solving Inequalities example 1 image 3

This leaves x on the left side of the inequality sign and 5 on the right side.

3Write your solution with the inequality symbol.

Solving Inequalities example 1 image 4

Any value less than 5 satisfies the inequality.

Example 2: solving linear inequalities

Solve:

Solving Inequalities example 2 image 1

In this case you need to add 4 to both sides.


Solving Inequalities example 2 image 2


This leaves 5x on the left side of the inequality sign and 30 on the right side.

In this case you need to divide both sides by 5.


Solving Inequalities example 2 image 3


This leaves x on the left side of the inequality sign and 6 on the right side.

Solving Inequalities example 2 image 4


Any value greater than or equal to 6 satisfies the inequality.

Example 3: solving linear inequalities with parentheses

Solve:

Solving Inequalities example 3 image 1

Let’s start by expanding the parentheses.


Solving Inequalities example 3 image 2


Then you need to add 12 to both sides.


Solving Inequalities example 3 image 3


This leaves 3x on the left side of the inequality sign and 24 on the right side.

In this case you need to divide both sides by 3.


Solving Inequalities example 3 image 4


This leaves x on the left side of the inequality sign and 8 on the right side.

Solving Inequalities example 3 image 5


Any value less than or equal to 8 satisfies the inequality.

Example 4: solving linear inequalities with unknowns on both sides

Solve:

Solving Inequalities example 4 image 1

In this case you need to subtract 2x from both sides.


Solving Inequalities example 4 image 2


This leaves 3x-6 on the left side of the inequality sign and 15 on the right side.

In this case you need to add 6 to both sides.


Solving Inequalities example 4 image 3


This leaves 3x on the left side of the inequality sign and 21 on the right side.

In this case you need to divide both sides by 3.


Solving Inequalities example 4 image 4


This leaves x on the left side of the inequality sign and 7 on the right side.

Solving Inequalities example 4 image 5


Any value greater than 7 satisfies the inequality.

Example 5: solving linear inequalities with fractions

Solve:

Solving Inequalities example 5 image 1

In this case you need to multiply both sides by 5.


Solving Inequalities example 5 image 2

In this case you need to subtract 3 from both sides.


Solving Inequalities example 5 image 3

Solving Inequalities example 5 image 4


Any value less than 7 satisfies the inequality.

Example 6: solving linear inequalities with non-integer solutions

Solve:

Solving Inequalities example 6 image 1

In this case you need to subtract 6 from both sides.


Solving Inequalities example 6 image 2

In this case you need to divide both sides by 6.


Solving Inequalities example 6 image 3


This can be simplified to \, \cfrac{1}{2} \, or the decimal equivalent.

Solving Inequalities example 6 image 4


Any value greater than or equal to \, \cfrac{1}{2} \, satisfies the inequality.

Example 7: solving linear inequalities and representing solutions on a number line

Represent the solution on a number line:

Solving Inequalities example 7 image 1

In this case you need to add 7 to both sides.


Solving Inequalities example 7 image 2

In this case you need to divide both sides by 2.


Solving Inequalities example 7 image 3

Solving Inequalities example 7 image 4


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.


Solving Inequalities example 7 image 5

Example 8: solving linear inequalities with negative x coefficients

Solve:

Solving Inequalities example 8 image 1

In this case you need to subtract 1 from both sides.


Solving Inequalities example 8 image 2

In this case you need to divide both sides by negative 2.


6 \div-2=-3

Because you divided by a negative number, you also need to change the direction of the inequality sign.


Solving Inequalities example 8 image 3

Example 9: solving linear inequalities and listing integer values that satisfy the inequality

List the integer values that satisfy:

Solving Inequalities example 9 image 1

In this case you need to subtract 1 from each part.


Solving Inequalities example 9 image 2

Solving Inequalities example 9 image 3


2 is not included in the solution set. 7 is included in the solution set. The integers that satisfy this inequality are:


Solving Inequalities example 9 image 4

Example 10: solving linear inequalities and listing integer values that satisfy the inequality

List the integer values that satisfy:

Solving Inequalities example 10 image 1

In this case you need to divide each part by 4.


Solving Inequalities example 10 image 2

Solving Inequalities example 10 image 3


\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:


Solving Inequalities example 10 image 4

Example 11: solving linear inequalities and representing the solution on a number line

List the integer values that satisfy:

Solving Inequalities example 11 image 1

In this case you need to subtract 5 from each part.


Solving Inequalities example 11 image 2

Solving Inequalities example 11 image 3

Solving Inequalities example 11 image 4


-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.


Solving Inequalities example 11 image 5

Teaching tips for solving inequalities

  • Students should master solving simple inequalities (one-step inequalities) before moving on to two-step inequalities (or multi-step inequalities).

  • Foster student engagement by practicing solving inequalities through classroom games such as BINGO rather than daily worksheets.

  • Student practice problems should have a variety of inequalities such as inequalities with fractions, negative numbers, and parentheses. (see examples above)

Easy mistakes to make

  • Solutions as inequalities
    Not including the inequality symbol in the solution is a common mistake. An inequality has a range of values that satisfy it rather than a unique solution so the inequality symbol is essential.

    For example, when solving x + 3 < 7 giving a solution of 4 or x = 4 is incorrect, the answer must be written as an inequality x < 4 .

  • Balancing inequalities
    Errors can be made with solving equations and inequalities by not applying inverse operations or not balancing the inequalities. Work should be shown step-by-step with the inverse operations applied to both sides of the inequality.

    For example, when solving x + 3 < 7 , adding 3 to both sides rather than subtracting 3 from both sides.

Practice solving inequalities questions

1. Solve:

 

Solving Inequalities prac question 1 image 1

3x < 24
GCSE Quiz False

x > 8
GCSE Quiz False

8
GCSE Quiz False

x<8
GCSE Quiz True

Solving Inequalities prac question 1 image 2

2. Solve:

 

Solving Inequalities prac question 2 image 1

x < 7
GCSE Quiz False

x \geq 7
GCSE Quiz True

x > 7
GCSE Quiz False

7
GCSE Quiz False

Solving Inequalities prac question 2 image 2

3. Solve:

 

Solving Inequalities prac question 3 image 1

x < 9
GCSE Quiz False

x \leq-1
GCSE Quiz False

x \leq9
GCSE Quiz True

9
GCSE Quiz False

Solving Inequalities prac question 3 image 2

4. Solve:

 

Solving Inequalities prac question 4 image 1

x > 3
GCSE Quiz True

x < 3.5
GCSE Quiz False

x \leq 3
GCSE Quiz False

3
GCSE Quiz False

Solving Inequalities prac question 4 image 2

5. Solve:

 

Solving Inequalities prac question 5 image 1

x < 4
GCSE Quiz False

x > 16
GCSE Quiz True

x\leq3
GCSE Quiz False

16
GCSE Quiz False

Solving Inequalities prac question 5 image 2

6. Solve:

 

Solving Inequalities prac question 6 image 1

x\geq 4
GCSE Quiz False

x\leq4
GCSE Quiz False

\cfrac{1}{4}
GCSE Quiz False

x\geq \, \cfrac{1}{4}
GCSE Quiz True

Solving Inequalities prac question 6 image 2

7. Represent the solution on a number line.

 

Solving Inequalities prac question 7 image 1

Solving Inequalities prac question 7 image 2

GCSE Quiz False

Solving Inequalities prac question 7 image 3

GCSE Quiz False

Solving Inequalities prac question 7 image 4

GCSE Quiz True

Solving Inequalities prac question 7 image 5

GCSE Quiz False

Solving Inequalities prac question 7 image 6

 

An open circle is required and all values less than 6 indicated.

8. Solve:

 

2-3x > 14

x > 4
GCSE Quiz False

x < -4
GCSE Quiz True

-x > 4
GCSE Quiz False

x < \, \cfrac{16}{3}
GCSE Quiz False

Solving Inequalities prac question 8

 

Change the direction of the inequality sign as you have divided by a negative number.

9. List the integer values that satisfy:

 

Solving Inequalities prac question 9 image 1

0, 1, 2
GCSE Quiz True

2, 3, 4, 5
GCSE Quiz False

3, 4, 5
GCSE Quiz False

-1, 0, 1, 2
GCSE Quiz False

Solving Inequalities prac question 9 image 2

 

-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:

 

Solving Inequalities prac question 10 image 1

1, 2, 3, 4, 5, 6, 7
GCSE Quiz False

2, 3, 4, 5, 6
GCSE Quiz False

1, 2, 3, 4, 5, 6
GCSE Quiz False

2, 3, 4, 5, 6, 7
GCSE Quiz True

Solving Inequalities prac question 10 image 2

 

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:

 

Solving Inequalities prac question 11 image 1

-2, -1, 0, 1
GCSE Quiz False

-1, 0
GCSE Quiz False

-1, 0, 1
GCSE Quiz True

-3, -2, -1, 0, 1, 2, 3, 4, 5
GCSE Quiz False

Solving Inequalities prac question 11 image 2

 

-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 FAQs

What is solving inequalities?

Solving inequalities is where you calculate the values that an unknown variable can be in an inequality.

How do you solve 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.

How does solving inequalities differ from solving equations?

Solving inequalities is similar to solving equations, but where an equation has one unique solution, an inequality has a range of solutions.

The next lessons are

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