Inequalities and Graphs of Inequalities
Linear inequalities are expressions in which the symbols of inequality, “,, < or>”, are used to compare any two values.
These could be numerals, algebraic terms, or a mix of both. Given below is a list of inequality symbols.
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:
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.
Learn More About Inequalities in This Video
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
\(x<\frac{35}{4}\)
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.
\(x<\frac{22-9}{11}\)
\(x<\frac{13}{11}\)
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.
\(x\geq\frac{20}{21}\)
Example 4:
\(7x\le5\)
Solution:
\(7x\times\frac{1}{7}\le5\times\frac{1}{7}\) Multiply \(\frac{1}{7}\) on both sides and simplify.
\(x\le\frac{5}{7}\)
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.
Solution:
Let p be the price of the chocolate bar.
If this is the case, the inequality can be expressed as follows:
p > 5
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.