Linear Inequalities

Linear inequalities are the expressions where any two values are compared by the inequality symbols such as, ‘<’, ‘>’, ‘≤’ or ‘≥’. These values could be numerical or algebraic or a combination of both. For example, 10<11, 20>17 are examples of numerical inequalities, and x>y, y<19-x, x ≥ z > 11 are examples of algebraic inequalities (also called literal inequalities). Learn linear inequalities for class 11 here. In this article, we are going to learn what is inequality in Math, linear inequalities, graphing of linear inequalities, and examples in detail.

What is Linear Inequality in Maths?

In Mathematics, inequality represents the mathematical expression in which both sides are not equal. If the relationship makes the non-equal comparison between two expressions or two numbers, then it is known as inequality in Maths. In this case, the equal sign “=” in the expression is replaced by any of the inequality symbols such as greater than symbol (>), less than symbol (<), greater than or equal to symbol (≥), less than or equal to symbol (≤) or not equal to symbol (≠). The different types of inequalities in Maths are polynomial inequality, rational inequality, absolute value inequality.

The symbols ‘<‘ and ‘>’ express the strict inequalities and the symbols ‘≤’ and ‘≥’ denote slack inequalities. A linear inequality seems exactly like a linear equation but there is a change in the symbol that relates two expressions.

How to Solve Inequalities in Maths?

Follow the below steps to solve all types of inequalities:

Step 1: First, write the inequality as an equation.

Step 2: Solve the given equation for one or more values.

Step 3: Now, represent all the values obtained in the number line.

Step 4: Use open circles to represent the excluded values on the number line.

Step 5: Find the interval.

Step 6: Now take any random value from the interval and substitute it in the inequality equation to check whether the values satisfy the inequality equation.

Step 7: Intervals that satisfy the inequality equation are the solutions of the given inequality equation.

Linear Inequalities Graphing

When we plot the graph for inequalities, we can see the graph of an ordinary linear function. But in the case of a linear function, the graph is a line and in the case of inequalities, the graph is the area of the coordinate plane that satisfies the inequality.

The linear inequality graph divides the coordinate plane into two parts by a borderline. This line is the line that belongs to the function. One part of the borderline consists of all solutions to the inequality. The borderline is dashed for inequalities ‘>’ and ‘<’ and solid for ‘≥’ and ‘≤’.

To plot an inequality graph, we need to follow three steps:

• Rearrange the given equation such that the variable ‘y’ is on the left-hand side and the rest of the equation on the right-hand side. For example, y>x+2
• Plot the graph for “y=x+2” line by putting the values for x.
• Remember to draw a solid line for y≤ or y≥ and a dashed line for y< or y>.
• Now, shade the line as per inequalities, such as above the line for a “greater than” (y> or y≥) and below the line for a “less than” (y< or y≤).

Also, read:

• Graphing Of Linear Equations
• Linear Inequalities In Two Variables

System of Linear Inequalities

A system of linear inequalities in two variables includes at least two linear inequalities in the identical variables. When we solve linear inequality then we get an ordered pair. So basically, in a system, the solution to all inequalities and the graph of the linear inequality is the graph displaying all solutions of the system. Let us see an example to understand it.

Example: Graph the Linear inequality: 2x – y >1, x – 2y < – 1

Solution: Given two inequality expressions:

2x – y >1   ….(i)

x – 2y < – 1  …(ii)

We need to find the solution for the given inequalities using a graph.

Let us take first, 2x – y > 1

We have to find the points to plot in the graph.

Putting x = 0 in equation (i) and putting ‘=’ instead of ‘>’

y = -1

Again putting y = 0 in equation (i) and putting ‘=’ instead of ‘>’

x = 1/2 = =0.5

Thus, the points are (0,-1) and (0.5,0).

If we put x=0 and y=0, to check for origin, then;

0 >1, which is not true.

Thus, the origin does not lie in the solution of 2x – y > 1.

Repeat the same step for equation (ii). Thus the required points will (0, 0.5) and (-1, 0).

Checking for origin, put x =0 and y = 0.

0 < -1, which is not true.

Thus, the origin does not lie in the solution of x – 2y < – 1.

Hence, the required solution in the graph will be:

Video Lesson

Solving Inequalities

Linear Inequalities Problems

Example 1:

Solve the inequality 4 ( x + 2 ) − 1 > 5 − 7 ( 4 − x )

Solution:

Given,

4 ( x + 2 ) − 1 > 5 − 7 ( 4 − x )

Expanding the brackets and multiplying by each term we get;

4 x + 8 − 1 > 5 − 28 + 7 x

4 x + 7 > − 23 + 7 x

Subtract 7 on both the sides

4x + 7 – 7 > -23 + 7x – 7

4x > -30 + 7x

Subtracting 7x from both the sides

4x – 7x > -30 + 7x – 7x

− 3 x > − 30

Multiplying both the sides by -1, the inequality gets reversed;

-3x (-1) < -30 x (-1)

3x < 30

Dividing both the sides by 3, we get;

3x/3 < 30/3

x < 10

Hence, x lies between -∞ and less than 10.

Example 2:

Solve the following inequality 8 ≤ 3 − 5 x < 12

Solution:

Given,

8 ≤ 3 − 5 x < 12

We can distribute the intervals as:

8 ≤ 3 – 5x and 3 – 5x < 12

Now taking one at a time.

8 ≤ 3 – 5x

Subtracting 3 on both the side

8 – 3 ≤ 3 – 5x – 3

5 ≤ -5x

Multiplying by (-1) on both sides reverses the inequality sign.

5 (-1) ≥ -5x (-1)

-5 ≥ 5x

Dividing both the sides by 5, we get;

-5/5 ≥ 5x/5

-1 ≥ x  … (i)

Now, taking the second interval, we have;

3 – 5x < 12

Subtracting 3 on both the sides;

3 – 5x – 3 < 12 – 3

-5x < 9

Multiplying by (-1) on both sides reverses the inequality sign.

(-1) (-5x) > 9 (-1)

5x > -9

Dividing by 5 on both the sides;

5x/5 > -9/5

x > -9/5  … (ii)

On combining both the intervals (i) and (ii), we can write;

− 1 ≥ x > − 9/5

or

-9/5 < x ≤ -1

Hence, x lies between the interval (-9/5,-1).

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