# Inequalities

## Summary

• An inequality basically compares two statements with different values.
• An inequality is written with either a greater than sign ( >) or less than sign (< ).
• If we divide or multiply by a negative number, the direction of the inequality is changed.
• To solve a quadratic inequality we need to carry out the following steps:

i) Set the inequality to zero
ii) Factor the equation
iii) Make intervals between the points found.
iv) Find the sign of the equation on the interval
v) Check which interval satisfies the inequality

• Also remember these signs:  $\ge$  greater than or equal to,
$\le$  less than or equal to

#### What are inequalities?

An inequality basically compares two statements with different values. We come across lots of inequalities every day in our lives. Some of the examples include speed limits on the highway (i.e  $\le$ 60 miles per hour, number of messages you can send each month from your cell phone (i.e  $\le$ 1000), height restriction on rides in adventure parks (i.e 4 $\ge$ feet).

All of these can be represented as mathematical inequalities. Now that you have an idea about what an inequality is we can move on and study how we can solve inequalities.

An inequality is written with either a greater than sign ( >) or less than sign (< ). These signs are what differentiates an inequality from an equation.

For example:  2x + 5 = 26 is an equation whereas 2x + 5 < 26 is an inequality.

You should know what the following symbols mean when dealing with inequalities:

$a\quad >\quad b$    a is greater than b

$a\quad <\quad b$   a is less than b

$a\quad \ge \quad b$   a is greater than or equal to b

$a\quad \le\quad b$   a is less than or equal to b

Inequalities are solved in the same way as equations are solved. For example if we add, subtract, multiply or divide on one side of the inequality, we do the same to the other side as well.

However, there is an exception when multiplying or dividing by a negative quantity. If we divide or multiply by a negative number, the direction of the inequality is changed.

#### Example#1

If we are given an inequality   $-\quad 2x\quad +\quad 4\quad >\quad 8$

We carry out the following steps to solve this:

$-\quad 2x\quad >\quad 8\quad -\quad 4$

$-\quad 2x\quad >\quad 4$

Now in the next step we divide 4 by -2 , so now we will swap the greater than sign (>) with the less than sign (< ).

$x\quad <\quad -\frac { 4 }{ 2 } \quad \Rightarrow \quad x\quad <\quad -2$         Ans

The above example consisted of a linear inequality. All linear inequalities are solved through the same principle and the only thing we must remember is the changing of the sign when multiplying or dividing the inequality with a negative number.

Next let’s study solving quadratic inequalities. In order to solve a quadratic inequalities we need to carry out the following steps:

1) Set the inequality to zero by taking terms consisting of the variable to one side of the equation.
2) Factor the equation and find the values of the variables where the equation becomes zero.
3) Make a number line with these values as the only points. Make intervals between these points.
4) Find the sign of the factors. Then find the sign of the equation on the interval.
5) The intervals that satisfy the inequality will be the result.

Let’s do an example now so you can see how these steps are applied.

#### Example #2

Q. Find the intervals that satisfy the relationship   $2{ x }^{ 2 }\quad -\quad 7x\quad >\quad -3$

Solution:

1) Set the inequality to zero:

$2{ x }^{ 2 }\quad -\quad 7x\quad +\quad 3\quad >\quad 0$

$2{ x }^{ 2 }\quad -\quad 7x\quad +\quad 3\quad =\quad 0$

2) Factor the above equation:

$(2x\quad -\quad 1)(x\quad -\quad 3)\quad =\quad 0$

Hence, we now know the zeros of the equation:

$x\quad =\quad 3\quad and\quad x\quad =\quad \frac { 1 }{ 2 }$

3) Now make a number line pointing these values and show the three regions:

4) Find the signs of the factors in each range:

 x\quad <\quad \frac { 1 }{ 2 } [/latex] \frac { 1 }{ 2 } \quad >\quad x\quad <\quad 3[/latex] $x\quad >\quad 3$$x\quad >\quad 3$ (2x - 1) negative positive positive (x - 3) negative negative positive (2x - 1)(x - 3) positive negative positive

5) Finally we can clearly see the range that satisfies the inequality:

$x\quad <\quad \frac { 1 }{ 2 } \quad and\quad x\quad >\quad 3$