Last modified on May 11th, 2024

chapter outline

 

Inequalities

Inequality is the mathematical symbol used to compare two values or expressions that are not equal. 

Symbols

Here are the four inequality notations or symbols used to write mathematical statements:

SymbolsWordsExamples
<Less than5x < 2
>Greater than7x > 16
Less than or equal tox – 7 ≤ 24
Greater than or equal tox + 7 ≥ 12
Inequality Symbols

Strict Inequality

The symbols < and > are known as strict inequalities since the expression on the left of the symbol must be less/greater than the expression on the right.

From the above table, 5x < 2 and 7x > 16 are the strict inequalities.

Slack Inequality

The symbols ≤ and ≥ are slack (weak) inequalities since the expression on the left of the symbols may be equal or less/greater than the expression on the right.

From the above table, the slack inequalities are x – 7 ≤ 24 and x + 7 ≥ 12.

Moreover, if two or more symbols are present in an expression, they are compound inequalities (sometimes, double inequalities).

Properties

Transitive 

It states that if one quantity is less than another, and the second quantity is less than a third quantity, then the first quantity is also less than the third quantity.

Mathematically, if a < b and b < c, then a < c

Also, we conclude that if a > b and b > c, then a > c

For example, if Marco is younger than Jacob, and Jacob is younger than Lucas, then Marco must be younger than Lucas.

Addition  

It states that adding the same quantity on both sides does not change the direction of the inequality.

Mathematically, 

If a > b, then a + c > b + c

If a < b, then a + c < b + c

For example, if Marco is younger than Jacob, then after 5 years, Marco will still be younger than Jacob. 

Subtraction  

It states that subtracting the same quantity from both sides does not change the direction of the inequality.

Mathematically,

If a > b, then a – c > b – c

If a < b, then a – c < b – c

For example, if Jacob is younger than Lucas, then 5 years ago, Jacob was still younger than Lucas.

Multiplication  

It states that multiplying both sides by the same positive quantity does not change the direction of the inequality.

Mathematically,

If a > b and c > 0, then ac > bc

If a < b and c > 0, then ac < bc

For example, Marco is younger than Lucas. If they double their ages, Marco will still be younger than Lucas.

However, multiplying both sides by the same negative quantity flips the direction of the inequalities, which means that:

If a > b and c < 0, then ac < bc

Also, if a < b and c < 0, then ac > bc

Division  

It states that dividing both sides by the same positive quantity does not change the direction of the inequality.

Mathematically,

If a > b and c > 0, then ${\dfrac{a}{c} >\dfrac{b}{c}}$

If a < b and c > 0, then ${\dfrac{a}{c} <\dfrac{b}{c}}$

However, dividing both sides by the same negative quantity flips the direction of the inequalities, which means:

If a > b and c < 0, then ${\dfrac{a}{c} <\dfrac{b}{c}}$

If a < b and c < 0, then ${\dfrac{a}{c} >\dfrac{b}{c}}$

For example, if x + 5 > -2, then on dividing both sides by (-1), we get

${\dfrac{x+5}{-1} <\dfrac{-2}{-1}}$

⇒ ${-\left( x+5\right) <2}$

Inversion 

It states that taking the inverse of both sides changes the direction of the inequality. 

Mathematically, 

If a > b, then ${\dfrac{1}{a} <\dfrac{1}{b}}$

If a < b, then ${\dfrac{1}{a} >\dfrac{1}{b}}$

For example, if 5 > 2, then ${\dfrac{1}{5} <\dfrac{1}{2}}$

However, if a > b, then -a < -b, and if a < b, then -a > -b

For example, since the numbers on the left are the smaller numbers on the number line.

Thus, from 2 to 9 is an increase, but from -2 to -9 is a decrease, as shown.

Inequalities

Solving

We solve inequalities to find the value of an unknown variable in an expression. If the variable is already independent, solving those basic inequalities is unnecessary. 

For example, x ≤ -11, y < 2, and z > 21 are the ‘solved’ inequalities.

Now, let us solve the inequality 3x – 7 < 2 + 8x

On subtracting 3x from both sides (subtraction property), we get

3x – 7 – 3x < 2 + 8x – 3x

⇒ -7 < 2 + 5x

On subtracting 2 from both sides (subtraction property), we get

-7 – 2 < 2 + 5x – 2

⇒ -9 < 5x

On dividing both sides by 5 (division property), we get

${\dfrac{-9}{5} <\dfrac{5x}{5}}$

⇒ ${\dfrac{-9}{5} <x}$

⇒ ${x >\dfrac{-9}{5}}$

Simplify ${\dfrac{a+5}{-3}\leq a+2}$

Solution:

Here, ${\dfrac{a+5}{-3}\leq a+2}$
Using multiplication property, we get
${\left( -3\right) \left( \dfrac{a+5}{-3}\right) \geq \left( -3\right) \left( a+2\right)}$
⇒ ${\left( a+5\right) \geq \left( -3a-6\right)}$
Using addition property, we get
${a+5+3a\geq -3a-6+3a}$
⇒ ${4a+5\geq -6}$
Using the subtraction property, we get
${4a+5-5\geq -6-5}$
⇒ ${4a\geq -11}$
Using the division property, we get
${\dfrac{4a}{4}\geq \dfrac{-11}{4}}$
⇒ ${a\geq \dfrac{-11}{4}}$
Thus, ${a\geq \dfrac{-11}{4}}$

Graphing

For One Variable

To plot the inequalities with one variable, we use a number line. Now, graphing the inequalities ≤ -8 and b > -5, we get:

Solve The Inequality

We observe that the solutions of inequalities are the rays graphed on the number line. Since the variables are already independent, these inequalities can be considered ‘solved.’

Here, 

  • The closed circle denotes that ‘a’ can equal -8. Thus, for slack inequalities (≤ and ≥), a closed dot indicates the endpoint to be a part of the solution. 
  • The open circle denotes that ‘b’ is not equal to -5. Thus, for the strict inequalities (< or >), an open dot indicates the endpoint is not part of the solution.

For Two Variables

Let us plot the inequality 3x + 1 > y on the graph.

First, we plot y = 3x + 1, as shown.

Graph The Inequality Step 1

Now, shading the area, we get

Graph Inequalities Step 2

Here, the dashed line is formed since 3x + 1 > y (‘less than’ symbol). However, we make it a solid line for ‘y ≤’ or ‘≥ y’.

Now, let us solve the inequality 5x – 4 < 0 very easily using graphs.

First, we sketch a graph of y = 5x – 4 as shown.

Inequality Graph Step 1

We observe that the line has a slope of 5 and an intercept on the y-axis of 4. If y = 0, x = ${\dfrac{4}{5}}$

By finding the values of x for which 2x + 3 is less than zero, we get those points on the graph where y is less than zero, which corresponds to the values of x less than ${\dfrac{4}{5}}$. 

This is the solution of the inequality. We mark this range on the graph, using the x-axis as the number line.

How to Graph an Inequality Step 2

Plot the graph of the inequality ${2y-\dfrac{x}{2}\leq 2}$

Solution:

Here, ${2y-\dfrac{x}{2}\leq 2}$
⇒ ${4y-x\leq 4}$ (by multiplication property)
⇒ ${4y\leq x+4}$ (by addition property)
⇒ ${y\leq \dfrac{x}{4}+1}$ (by division property)
Since the given one is a weak inequality, a solid line is formed. Now, shading the area of the graph ${y=\dfrac{x}{4}+1}$, we get the solution as shown.

Last modified on May 11th, 2024