Solving Inequalities

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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.

  1. ax + by ≤ c

  2. ax + by < c

  3. ax + by > c

  4. 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.

  1. 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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  1. 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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  1. 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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  1.  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

  1. You can solve simple inequalities by adding, subtracting, multiplying or dividing both sides until you are left with the variable on its own.

  2. 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.

  1. 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.

FAQ (Frequently Asked Questions)

Question 1) How do we Solve Inequalities?

Answer)This is how to solve inequalities -

  1. You can solve simple inequalities by adding, subtracting, multiplying or dividing both sides until you are left with the variable on its own.

  2. 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.

  1. You don't need to multiply or divide by a variable (unless you know that it is always positive or always negative).

Question 2)When you Divide an Inequality by a Negative?What is the Rule for Flipping Inequality Signs?

Answer)There is one very important exception to the rule that multiplying or dividing an inequality is the same as multiplying or dividing an equation. Whenever you multiply or divide any given inequality by any negative number, you must be able to flip the inequality sign.


There is one very important exception to the rule that multiplying or dividing an inequality is similar to multiplying or dividing an equation. Now, whenever you multiply or divide an inequality by a negative number, you must flip the inequality sign.