Sometimes we need to solve Inequalities like these:
Our aim is to have x (or whatever the variable is) on its own on the left of the inequality sign:
We call that "solved".
Example: x + 2 > 12
Subtract 2 from both sides:
x + 2 − 2 > 12 − 2
x > 10
How to Solve
Solving inequalities is very like solving equations , we do most of the same things ...
... but we must also pay attention to the direction of the inequality .
Some things can change the direction !
< becomes >
> becomes <
≤ becomes ≥
≥ becomes ≤
Safe Things To Do
These things do not affect the direction of the inequality:
- Add (or subtract) a number from both sides
- Multiply (or divide) both sides by a positive number
- Simplify a side
Example: 3x < 7+3
We can simplify 7+3 without affecting the inequality:
But these things do change the direction of the inequality ("<" becomes ">" for example):
- Multiply (or divide) both sides by a negative number
- Swapping left and right hand sides
Example: 2y+7 < 12
When we swap the left and right hand sides, we must also change the direction of the inequality :
12 > 2y+7
Here are the details:
Adding or Subtracting a Value
We can often solve inequalities by adding (or subtracting) a number from both sides (just as in Introduction to Algebra ), like this:
Example: x + 3 < 7
If we subtract 3 from both sides, we get:
x + 3 − 3 < 7 − 3
And that is our solution: x < 4
In other words, x can be any value less than 4.
What did we do?
And that works well for adding and subtracting , because if we add (or subtract) the same amount from both sides, it does not affect the inequality
Example: Alex has more coins than Billy. If both Alex and Billy get three more coins each, Alex will still have more coins than Billy.
What If I Solve It, But "x" Is On The Right?
No matter, just swap sides, but reverse the sign so it still "points at" the correct value!
Example: 12 < x + 5
If we subtract 5 from both sides, we get:
12 − 5 < x + 5 − 5
That is a solution!
But it is normal to put "x" on the left hand side ...
... so let us flip sides (and the inequality sign!):
Do you see how the inequality sign still "points at" the smaller value (7) ?
And that is our solution: x > 7
Note: "x" can be on the right, but people usually like to see it on the left hand side.
Multiplying or Dividing by a Value
Another thing we do is multiply or divide both sides by a value (just as in Algebra - Multiplying ).
But we need to be a bit more careful (as you will see).
Everything is fine if we want to multiply or divide by a positive number :
Example: 3y < 15
If we divide both sides by 3 we get:
3y /3 < 15 /3
And that is our solution: y < 5
Well, just look at the number line!
For example, from 3 to 7 is an increase , but from −3 to −7 is a decrease.
See how the inequality sign reverses (from < to >) ?
Let us try an example:
Example: −2y < −8
Let us divide both sides by −2 ... and reverse the inequality !
−2y < −8
−2y /−2 > −8 /−2
And that is the correct solution: y > 4
(Note that I reversed the inequality on the same line I divided by the negative number.)
So, just remember:
When multiplying or dividing by a negative number, reverse the inequality
Multiplying or Dividing by Variables
Here is another (tricky!) example:
Example: bx < 3b
It seems easy just to divide both sides by b , which gives us:
... but wait ... if b is negative we need to reverse the inequality like this:
But we don't know if b is positive or negative, so we can't answer this one !
To help you understand, imagine replacing b with 1 or −1 in the example of bx < 3b :
- if b is 1 , then the answer is x < 3
- but if b is −1 , then we are solving −x < −3 , and the answer is x > 3
The answer could be x < 3 or x > 3 and we can't choose because we don't know b .
Do not try dividing by a variable to solve an inequality (unless you know the variable is always positive, or always negative).
A Bigger Example
Example: x−3 2 < −5.
First, let us clear out the "/2" by multiplying both sides by 2.
Because we are multiplying by a positive number, the inequalities will not change.
x−3 2 ×2 < −5 ×2
x−3 < −10
Now add 3 to both sides:
x−3 + 3 < −10 + 3
And that is our solution: x < −7
Two Inequalities At Once!
How do we solve something with two inequalities at once?
Example: −2 < 6−2x 3 < 4
First, let us clear out the "/3" by multiplying each part by 3.
Because we are multiplying by a positive number, the inequalities don't change:
−6 < 6−2x < 12
−12 < −2x < 6
Now divide each part by 2 (a positive number, so again the inequalities don't change):
−6 < −x < 3
Now multiply each part by −1. Because we are multiplying by a negative number, the inequalities change direction .
6 > x > −3
And that is the solution!
But to be neat it is better to have the smaller number on the left, larger on the right. So let us swap them over (and make sure the inequalities point correctly):
−3 < x < 6
- Many simple inequalities can be solved by adding, subtracting, multiplying or dividing both sides until you are left with the variable on its own.
- Multiplying or dividing both sides by a negative number
- Don't multiply or divide by a variable (unless you know it is always positive or always negative)
- Join for FREE
- Printable Worksheets
- Online Lessons
- Test Maker™
- Printable Games
- Worksheet Generator
- Plans & Pricing
Printable & online resources for educators
- Test Maker TM
- Browse All Questions
- Questions With Images
- Advanced Search
Share/Like This Page
Filter by grade.
You are browsing Grade 9 questions. View questions in All Grades .
Grade 5 Grade 6 Grade 7 Grade 8 Grade 9 Grade 10 Grade 11 Grade 12
- All Subjects w/ Images (7037)
- By ELA/Literacy Standard
- By Math Standard
- All Subjects (25106)
- Arts (1038)
- English Language Arts (5953)
- English as a Second Language ESL (3250)
- Health and Medicine (901)
- Life Skills (397)
Arithmetic and Number Concepts
Function and algebra concepts, absolute value, algebraic expressions, complex numbers, direct and inverse variation, functions and relations, inequalities, linear equations, literal equations, nonlinear equations and functions, number properties, polynomials and rational expressions, quadratic equations and expressions, sequences and series, systems of equations, geometry and measurement, mathematical process, statistics and probability concepts.
- Physical Education (842)
- Science (7920)
- Social Studies (3991)
- Study Skills and Strategies (159)
- Technology (363)
- Vocational Education (816)
Ninth Grade (Grade 9) Inequalities Questions
You can create printable tests and worksheets from these Grade 9 Inequalities questions! Select one or more questions using the checkboxes above each question. Then click the add selected questions to a test button before moving to another page.
- [math]x > -5[/math]
- [math]x >= -2/3[/math]
- [math]x > 2/3[/math]
- [math]x > -3/2[/math]
- [math]x >= 3/2[/math]
- [math]x> -4[/math]
- [math]x > 4[/math]
- [math]x < -4[/math]
- [math]x < 4[/math]
- [math]x < -8[/math]
- [math]x > -8[/math]
- [math]x > 8[/math]
- [math]x < 8[/math]
- [math]x>=-2 1/3[/math]
- [math]x>=2 1/3[/math]
- none of the above
- [math]x < -3/4[/math]
- [math]x > -3/4[/math]
- [math]x < 3/4[/math]
- [math]x > 3/4[/math]
- [math]x < -3 [/math]
- [math]x < 3 [/math]
- [math]x > -3[/math]
- [math]x > 3[/math]
- x > -2/7
- x < -2/7
- x > -10/7
- x < -10/7
- [math]7< f <11[/math]
- [math]f<7 or f>11[/math]
- [math]f<=7 or f>=11[/math]
- FREE Printable Worksheets
- Common Core ELA Worksheets
- Common Core Math Worksheets
- Share on Facebook
- Tweet This Resource
- Pin This Resource
In this inequalities worksheet, 9th graders solve and complete 25 various types of problems. First, they determine the portion of the graph that should be shaded for each inequality. Then, students graph each system of inequalities. In addition, they determine which graph best represents each inequality.
Start Your Free Trial
Save time and discover engaging curriculum for your classroom. Reviewed and rated by trusted, credentialed teachers.
- Collection Types
- Activities & Projects
- Graphics & Images
- Handouts & References
- Lab Resources
- Learning Games
- Lesson Plans
- Primary Sources
- Printables & Templates
- Professional Documents
- Study Guides
- Instructional Videos
- Performance Tasks
- Graphic Organizers
- Writing Prompts
- Constructed Response Items
- AP Test Preps
- Lesson Planet Articles
- Online Courses
- Interactive Whiteboards
- Home Letters
- Unknown Types
- Stock Footages
- All Resource Types
See similar resources:
Graphing systems, do these systems meet your expectations, systems of equations and inequalities, systems of linear inequalities, graphical system of inequalities, solving systems of equations three ways, graphing systems of equations, richland com. college: graphing systems of inequalities, systems of equations, desmos graphing calculator.