2.6 Inequalities

Hi Everyone!

Today in class we learned about inequalities, which are statements that express that two quantities or expressions are not equal by using the signs...

(less than)

(greater than)

(less than or equal to)

(greater than or equal to)

Solving inequalities is similar to solving equations except for a few differences:

When you divide by a negative number, you must switch the inequalities sign:

Ex.

You must also switch the sign if you take the reciprocal of both expressions:

Ex.

Also, when you're solving an inequality, you can't square both sides or take the square root of both sides.

After you solve an inequality you must put your answer into interval notation:

You use parenthesis to represent less than/greater than, and use brackets to express less than or equal to/greater than or equal to:

Ex.

Interval Notation:

Ex.

Interval Notation:

Ex.

Interval Notation: (- infinity, 5)U(8, infinity)

*U=Union

You can also show your answers to inequalities as graphs, like the ones below:

open dot: less than/greater than

closed dot: less than or equal to/greater than or equal to

So those are basically the only differences between inequalities and equations so...Let's solve some inequalities!

Ex.

divide each side of the inequality by 5

Interval Notation:

Ex.

One way to solve an inequality like this is to split it into two inequalities:

and

solve both inequalities by subtracting 7 from each side and you get...

and

Interval Notation: (-infinity, -6) U (-3, infinity)

A faster way to solve this inequality is by solving both inequalities at the same time:

subtract 7 from the whole inequality...

Interval Notation: (-infinity, -6)U(-3, infinity)

Ex.

First, multiply the whole inequality by 2

Then, divide the inequality by 3

Interval Notation:

Ex.

You know that the left side of the inequality has to be greater than zero, so in order for it to be greater than zero, the ratio must be positive. So, the denominator has to be positive to make the inequality true. Therefore....

subtract 4 from each side

Interval Notation: (-4, infinity)

Absolute Value Inequalities:

Solving absolute value inequalities is basically the same as solving absolute value equations.

Ex.

split the inequality into two inequalities: one with x-3 and another with -(x-3)

and

then, solve the inequalities in the same way you would solve a normal inequality...

and

Interval Notation: (1,5)

So, that's it. I hope this is helpful.

-Olivia