We already know how to solve linear equations in one and two variables. Now, it is time to learn how to graph linear inequalities in one and two variables on a number line and coordinate axis. The relationship between variables in linear inequalities is represented by four inequality symbols. These inequality signs are:

  • <                     Less than
  • <                     Greater than
  • \leq           Less than or equal to
  • \geq          Greater than or equal to

We know how signs of the inequalities flip when we multiply them by a negative number. When we say that c>d then it means that c is strictly "greater" than d. The same goes for c<d, only "greater than" will be replaced by the "less than" word. When we say c \leq d, then it means that c is either "less than or equal to d". The same goes for c\geq d, only "less than" will be replaced by the "greater than".

c<d means d is on the right side and c is on the left side of the number line. For example, 3<7  is a true solution for the inequality c<d. If you say that c \leq d then 2<7 and 7 \leq 7 both are true solutions.

The above examples were related to simple linear inequalities. However, sometimes you are also given double inequalities, in which the inequality sign comes two times. For example, consider the following problem:

Write the following inequalities in the form of double inequality.

a>b and a<c

Before solving, we know that a, b and c are interconnected. We can simply write them as a single inequality by seeing which expression is repeated. "a" comes two times in the above example, so we will place it in the middle. Since b is lesser than a, so it will be positioned first and c will be at third position.

b<a<c

Let's see another example below.

x-1 \geq -2x+3 and -3x+1 \leq -2x+3

Now, first, consider which algebraic expression is repeated. -2x+3 comes two times in the above example, so it will be positioned in the middle of the double inequality. The final inequality will look like this:

x-1 \geq -2x+3 \geq -3x+1

Linear Inequality in One Variable

Linear equations in one variable contain a single variable only. We can donate linear inequality in one variable in interval as well as set builder notation. In interval notation, we either use parentheses or square brackets depending on the beginning and endpoint of the interval. While denoting a closed interval in interval notation, we use square brackets on both sides because both points are part of the solution set. The graph of the linear inequality is represented on a number line.

We will solve some of the examples of linear inequalities in one variable and write them in interval notation. We will also represent them on a number line.

Example 1

Solve the inequality 3x+2>2(x-1).

Solution

First, we will multiply the constant 2 with the expression x-1 inside the brackets on the right hand side of the inequality:

3x+2>2x-2

Now, subtract 2x from both sides of the inequality to get:

3x-2x+2>2x-2x-2

x+2>-2

Again subtracting 2 from sides of the inequality will yield:

x>-2-2

x>-4

Hence, the final solution is x>-4. It can be written in the interval notation as:

(-4,\infty)

You can see that the beginning and endpoints are not part of the solution set, so we have used parenthesis on both sides in the interval notation and open circle on -4 on the number line. The inequality on the number line will look like this:

Solution set on a number line

 

 

Example 2

Solve the inequality \frac{x+2}{3}+\frac{2x+4}{4} \geq \frac{x+5}{3}-\frac{x+2}{2}. Write the solution set in interval notation and represent it on a number line.

Solution

\frac{x+2}{3}+\frac{2x+4}{4} \geq \frac{x+5}{3}-\frac{x+2}{2}

Take L.C.M of 3 and 4 on the left hand side, and 3 and 2 on the right hand side of the inequality:

\frac{4(x+2)+3(2x+4)}{12} \geq \frac{2(x+5)-3(x+2)}{6}

Multiplying the terms inside the brackets with constants will result in the following expression:

\frac{4x+8+6x+12}{12} \geq \frac{2x+10-3x-6}{6}

\frac{10x+20}{12} \geq \frac{-x-4}{6}

Cross multiply the denominators with the expressions in the numerator on both sides of inequality:

60x+120 \geq -12x-48

Adding 12x and subtracting 120 from both sides will give:

72x \geq -72

Divide both sides by 72 to get the following answer:

x \geq -1

In the interval notation, we will write the answer like this:

[-1,\infty)

As the final answer x \geq -1 means x is greater than or equal to -1, so -1 is included in the interval. We have used a square bracket before -1 in the interval notation and filled circle on the number line to indicate that. On a number line the solution will look like this:

Solution set on a number line
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Linear Inequality in Two Variables

So far, we have discussed how to solve linear inequality in one variable. In this section, we will discuss linear inequality in two variables. Linear inequality in two variables involves two variables instead of one. They are solved in the same way as the linear equations in two variables.  Linear inequalities in two variables are represented like this:

Ax+By \geq C

When the values of x and y are substituted in the inequality, the final answer is a true statement. The solution of linear inequalities in two variables is represented as an ordered pair (x,y).

Example 1

Does the ordered pair (-3,4) satisfy the inequality -4x-2y>2?

Solution

To answer the question in this example, we will substitute the values in the ordered pair in the inequality. According to the question, x=-3 and y=4.

-4(-3)-2(4)>2

12-8>2

4>2

Hence, the ordered pair (-3,4) is the true solution to the inequality -4x-2y>2.

Example 2

Does the ordered pair (-1,5) satisfies the inequality 5x+3x \leq 11?

Solution

To answer this question, substitute the values from the ordered pair in the inequality 5x+3x \leq 11.

In the ordered pair (-1,5) x=-1 and y=5.

5(-1)+3(5) \leq 11

-5+15 \leq 11

10 \leq 11

Hence, the ordered pair (-1,5) satisfies the inequality 5x+3x \leq 11.

Graphing Linear Inequalities in Two Variables

We know that when linear equations are plotted in an xy plane, the graph is a straight line. The graph of linear inequalities is also a straight line, but it divides the coordinate plane into two parts. One part of the plane which is mostly shaded represents the inequality. Let us learn how to graph linear inequalities in a coordinate plane through the following examples.

Example 1

Graph the inequality y \leq 2x-2.

Solution

To graph the above inequality, first, we will construct a table as we do before graphing linear equations.

xy
-2-6
-1-4
0-2
10
22
34

Using the points on the table plot a graph and shade the region below the line because the inequality has less than equal to sign.

Graph of an inequality in two variables

Example 2

Solve the inequality y+5 \geq 3x+6.

Solution

Simplify the inequality before constructing a table. Subtract 5 from both sides of the inequality to get the following expression:

y+5-5 \geq 3x+6-5

y \geq 3x+1

Now, construct a table of the above linear inequality like this:

xy
-2-5
-1-2
01
14
27
310

Using the values of x and y from the table, plot the graph in the coordinate axis and highlight the region above the line because there is greater than equal to sign.

Graph of an inequality in two variables

The shaded region above the line represents a solution of the inequality.

So far, we have learned how to plot the graphs of the inequalities which have \leq and \geq signs. Inequalities involving \leq and \geq signs are plotted using a solid straight line. On the other hand, the inequalities which involve "<" or ">" sign are plotted using a dotted or dashed straight line because these are strict inequalities.

Now, we will see how to plot the graphs when the inequality has a "<" or ">" sign.

Example 3

Graph the inequality y<-x+3.

Solution

Before graphing, construct a table with the points x and y.

xy
-25
-14
03
12
21
30

Using the points from the table, plot the graph of the inequality like this:

An inequality graph in two variables

 

You can see that the dotted line is used to represent the straight line and the region below the dotted line is shaded.

Example 4

Graph the inequality y+9 >3x+16.

Solution

To construct the table, first solve the inequality by applying simple arithmetic operations.

Subtracting 9 from both sides of the inequality will give:

y+9-9>3x+16-9

y>3x+7

Construct the table of the inequality y>3x+7 like this:

xy
-21
-14
07
110
213
316

Using the above table, construct a graph like this:

Graph of an inequality in two variables

As the inequality was strict, i.e. it has greater than sign, therefore we used a dotted straight line to represent it.

 

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.

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