What is Graphing Inequalities? How to Graph Inequalities? – BYJUS

Inequalities and Graphs of Inequalities

We know that an equation states that two expressions are equal. Inequalities on the other hand, state that two expressions are not equal. We use the concept of inequalities to compare expressions that are not exactly equal to one another using symbols like (<, >, ≥, and ≤). We will learn how such inequalities are framed and how they can be represented on a number line....Read MoreRead Less

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Inequalities and Graphs of Inequalities

Linear inequalities are expressions in which the symbols of inequality, “,, < or>”, are used to compare any two values.

ineq

These could be numerals, algebraic terms, or a mix of both. Given below is a list of inequality symbols.

table

Examples:

Numeric inequalities:

10 < 19

20 > 9

Algebraic inequalities:

x > y

y < 31 – x, 

y ≥  z  > 19 

What is Linear inequality in Math?

An inequality is a mathematical expression whose two sides are not equal. In mathematics, inequality occurs when a non-equal comparison between two expressions or two numbers takes place. The symbols of inequality are:

greater than symbol (>), 

less than symbol (<), 

greater than or equal to symbol (\(\geq\)), 

less than or equal to symbol (\(\leq\)), 

or not equal to symbol (\(\neq\)). 

In the case of a linear inequality, the expressions on both the sides of the symbols of inequality are linear expressions (highest degree is 1). 

How to write an inequality?

To understand how this can be done, let us solve an example:

 

The entry for people to a water ride in an amusement park is based on a condition: the height of the person should be greater than or equal to 3.5 feet. Represent this in the form of an inequality. 

 

Answer: Understand the question and figure out the unknown quantity. A variable is assigned to this unknown quantity.


Variable

Inequality

Minimum height required.

Words

Let ‘h’ be the height of the people who can enter the water ride

greater than or equal to

3.5 feet

inequality

h

+3.5

How can we solve an Algebraic inequation?

An algebraic inequation can be solved as follows:

 

Step 1: Write the inequality as an equation first. 

Step 2: For one or more values, solve the given equation. 

Step 3: Rearrange the given equation such that the variable is on the left-hand side and the rest of the terms are on the right-hand side.

How can we represent an Algebraic inequation on a number line?

Let us consider an inequality: h +3.5.

Here is how we can represent the inequality on the number line:

line

It is important to note that for an inequality, the solutions are a range of values, as opposed to equations that have only one solution. The circle on which the number is placed denotes the start of the inequality. Here is how the inequality would have looked like in two different scenarios other than the one shown here.

line1

line2

Solved Graphing Inequalities Examples

Solve the given inequality and plot the inequality on a number line.

 

Example 1:

\(x-\frac{3}{4}<8\)

 

Solution:

\(x-\frac{3}{4}<8\)

 

\(x-\frac{3}{4}+\frac{3}{4}<8+\frac{3}{4}\)    Add \(\frac{3}{4}\) on both sides.

 

\(x<\frac{32+3}{4}\)                     Simplify

 

\(x<\frac{35}{4}\)

 

 

 

line3

 

 

 

Example 2:

\(x+\frac{9}{11}>2\)

 

Solution:

\(x+\frac{9}{11}>2\)

 

\(x+\frac{9}{11}-\frac{9}{11}>2-\frac{9}{11}\)     Subtract \(\frac{9}{11}\) from both sides.

 

 

\(x<2-\frac{9}{11}\)                       Simplify.

 

\(x<\frac{22-9}{11}\)

 

\(x<\frac{13}{11}\)

 

 

 

line4

 

 

 

Example 3:

\(7x+\frac{1}{3}\geq7\)

 

Solution:

\(7x+\frac{1}{3}-\frac{1}{3}\geq7-\frac{1}{3}\)     Subtract \(\frac{1}{3}\) from both sides.

 

\(7x\geq7-\frac{1}{3}\)                    Simplify.

 

\(7x\geq\frac{21-1}{3}\)

 

\(7x\times\frac{1}{7}\geq\frac{20}{3}\times\frac{1}{7}\)          Multiply \(\frac{1}{7}\) on both sides and simplify.

 

\(x\geq\frac{20}{21}\)

 

 

 

line5

 

 

 

Example 4:

\(7x\le5\)

 

Solution:

\(7x\times\frac{1}{7}\le5\times\frac{1}{7}\) Multiply \(\frac{1}{7}\) on both sides and simplify.

 

\(x\le\frac{5}{7}\)

 

 

 

line6

 

 

 

Example 5:

To get a decent profit by selling chocolates, Clarise has to sell each chocolate bar for more than $5. Express this situation in the form of an inequality and draw a graph for the same.

 

 

 

choco

 

 

 

Solution:

Let p be the price of the chocolate bar.

 

If this is the case, the inequality can be expressed as follows:

 

p > 5

 

 

 

line7

Frequently Asked Questions on Graphing Inequalities

An inequality is a conditional statement that is represented mathematically using numbers, variables, or both.

It can be plotted on a numberline. The point where the condition starts is encircled, and as per the question, the remaining section of the number line is shaded.