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
Symbol |
Words |
Example |
< |
This symbol is known as the less than symbol. |
7x < 25 |
> |
This symbol is known as the greater than symbol. |
2x >1 |
≥ |
This symbol is known as the greater than or equal to symbol. |
5 ≥ 2+1 |
≤ |
This symbol is known as the less than or equal to symbol. |
2 ≤ 4 |
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.
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ax + by ≤ c
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ax + by < c
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ax + by > c
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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.
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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.
(image will be uploaded soon)
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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.
uploaded soon)
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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.
uploaded soon)
Important Points you Need to Remember
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You can solve simple inequalities by adding, subtracting, multiplying or dividing both sides until you are left with the variable on its own.
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But these things will tend to change the direction of inequality:
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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.