How to Solve Linear Inequalities Step by Step with Rules and Graphs
Solving linear inequalities using the graphical method is an easy way to find the solutions for linear equations. Now to solve a linear equation in one variable is easy, where we can easily plot the value in a number line. But in the case of two-variable, we need to plot the graph in an x-y plane. A linear function is involved in solving linear inequalities. A mathematical expression that contains the symbol equal-to (=) is known to be an equation. The equality symbol basically signifies that the left-hand side of the expression is equal to the right-hand side of the expression. If two mathematical expressions contain such symbols ‘<’(less than symbol), ‘>’ (greater than symbol), ‘≤’(less than or equal to symbol) or ‘≥’ (greater than or equal to symbol), they are known as inequalities. In this article we are going to discuss what is an inequality equation,solving inequalities .
Let’s say for example,
Statement 1 – Jack is 20 years old.
The equality symbol can be mathematically expressed as x= 20.
Statement 2 – Now if I say Jack’s age is at least 20 years, then this can be expressed as x ≥ 20.
Thus, Statement 1 that is given above is an equation and Statement 2 is an inequality.
Sometimes we do Need to Solve 2 Inequalities These
Solving Linear Inequalities
We have already discussed what is an inequality equation. Let’s now discuss the method of solving 2 inequalities graphically. The graph of a linear equation is basically a straight line and any point in the Cartesian plane with respect to that will lie on either side of the line. Let us consider the expression ax + by for a linear equation in two variables. The following inequalities can be framed using the expression ax+by.
ax + by ≤ c
ax + by < c
ax + by > c
ax + by ≥ c
Linear Inequalities Graphing
For solving 2 inequalities that are mentioned above, we graph the linear expression and can make the following conclusions about the inequality.
ax + by < c
The region lying below the line ax + by = c or the region that is marked as II consists of all those points that will satisfy the inequality ax + by < c. The region II is known to be the solution region for the inequality of the type ax + by < c. The line is dotted since the solution region excludes the line ax + by = c.
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ax + by ≤ c
The region that lies below and includes the line ax + by = c or the region marked as II, it consists of all those points that will satisfy the inequality ax + by ≤ c. The region II is known to be the solution region for the inequality of the type ax + by ≤ c.
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ax + by > c
The region lying in the upper half of the line ax + by = c or the region marked as I and consists of all those points that would satisfy the inequality ax + by > c. The region I is known to be the solution region for the inequality of the type ax + by > c. Since the solution region excludes the line ax + by = c, therefore we say that the line is dotted.
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ax + by ≥ c
The region lying below and including the line ax + by = c or the region marked I consist of all those points that would satisfy the inequality ax + by ≥ c. The region I is known to be the solution region for the inequality of the type ax + by ≥ c.
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Important Points you Need to Remember
You can solve simple inequalities by adding, subtracting, multiplying or dividing both sides until you are left with the variable on its own.
But these things will tend to change the direction of inequality:
When we multiply or divide both the sides by a negative number.
When we swap the right hand sides and the left hand sides.
You don't need to multiply or divide by a variable (unless you know that it is always positive or always negative).
Questions to be Solved
Question 1) Solve for the value of x and check for : x + 5 = 3
Solution)Using the same procedures learned above, we subtract 5 from each side of the equation obtaining,
x+5-5 = 3-5
Therefore, the value of x = -2
Let’s check that the value we have got is correct or not.
Putting the value of x as -2 in the equation we have,
x+5 = 3
-2+5 = 3
3 = 3
Therefore, this proves that the value of x we have got is correct since both R.H.S and L.H.S are equal.
Question 2) Solve for the value of x and check for : x + 9 = 3
Solution)Using the same procedures learned above, we subtract 9 from each side of the equation obtaining,
x + 9 - 9 = 3 - 9
x + 0 = -6
Therefore, the value of x = -6
Let’s check that the value we have got is correct or not.
Putting the value of x as -6 in the equation we have,
x + 9 = 3
-6 + 9 = 3
3 = 3
Therefore, this proves that the value of x we have got is correct since both R.H.S and L.H.S are equal.
Question 3) Solve the following inequality -2(x+3)<10
Solution) Given inequality , 2(x+3)<10
Now first we need to divide both the sides by the number 2, we get;
= -(x+3) < 5
When we open the brackets we get,
= -x-3<5
Now we need to add 3 on both the sides,
= -x-3+3 < 5+3
=-x +0 < 8
Now divide both sides by -1 to convert the inequality into a positive one.
= -x /-1 < 8 /-1
We get , x>-8.
FAQs on Solving Inequalities with Methods and Examples
1. What is an inequality in maths?
An inequality is a mathematical statement that compares two expressions using symbols such as <, >, ≤, or ≥ instead of an equals sign. It shows that one quantity is greater than, less than, or not equal to another.
- < means less than
- > means greater than
- ≤ means less than or equal to
- ≥ means greater than or equal to
2. How do you solve a simple linear inequality?
To solve a linear inequality, isolate the variable using the same steps as solving an equation, but flip the sign if you multiply or divide by a negative number.
- Example: Solve 2x + 3 > 7
- Subtract 3: 2x > 4
- Divide by 2: x > 2
3. Why do you flip the inequality sign when multiplying by a negative?
You flip the inequality sign because multiplying or dividing both sides by a negative number reverses the order of values on the number line.
- Example: Start with 3 < 5
- Multiply both sides by −1: −3 and −5
- Since −3 is greater than −5, the inequality becomes −3 > −5
4. How do you solve inequalities with brackets?
To solve an inequality with brackets, expand the brackets first, then simplify and isolate the variable.
- Example: Solve 2(x − 3) ≤ 8
- Expand: 2x − 6 ≤ 8
- Add 6: 2x ≤ 14
- Divide by 2: x ≤ 7
5. How do you represent inequalities on a number line?
Inequalities are represented on a number line using open or closed circles and shading in the correct direction.
- Use a closed circle for ≤ or ≥ (value included)
- Use an open circle for < or > (value not included)
- Shade right for greater than, left for less than
6. What is the difference between an equation and an inequality?
An equation shows two expressions are equal, while an inequality shows one expression is greater or less than another.
- An equation uses = and usually has one or more exact solutions (e.g., x = 4)
- An inequality uses <, >, ≤, or ≥ and often has a range of solutions (e.g., x > 4)
7. How do you solve a compound inequality?
A compound inequality contains two inequalities joined by “and” or “or,” and each part must be solved carefully.
- Example (and): 2 < x ≤ 6 means x is between 2 and 6
- Example (or): x < −1 or x > 3 means values outside the interval
8. How do you solve quadratic inequalities?
To solve a quadratic inequality, find the roots of the related quadratic equation and test intervals to determine where the inequality is true.
- Example: Solve x² − 4 > 0
- Factor: (x − 2)(x + 2) = 0
- Roots: −2 and 2
- Solution: x < −2 or x > 2
9. Can inequalities have infinite solutions?
Yes, most inequalities have infinitely many solutions because they represent a range of values rather than a single number.
- Example: x > 1 includes 2, 3, 4, 1.5, and infinitely many more
- Solutions are often written in interval notation, such as (1, ∞)
10. What are common mistakes when solving inequalities?
The most common mistake when solving inequalities is forgetting to flip the sign when multiplying or dividing by a negative number.
- Not reversing the sign after multiplying/dividing by a negative
- Incorrectly graphing open vs closed circles
- Treating inequalities exactly like equations without checking direction
- Ignoring solution intervals in compound inequalities
