Solving and Graphing Inequalities

solving and graphing inequalities

with review

The crocodile eats the bigger number.

Remember solving equations by isolating the variable?

We solve inequalities the same way! There can be one-step or two-step or multi-step depending on how many operations there are. Just remember to isolate the variable by getting the variable on one side of the inequality and everything else to the other side.

There is only one big difference!!!

Look what happens when we multiply or divide by a negative to both sides; 4<6 true -4<-6 false. Since the crocodile needs to eat the bigger number, it will turn around and eat the -4, thus -4>-6. This is always true, so what we have to do when multiplying or dividing by a negative number is turn the crocodile around or switch the inequality to the opposite direction.

See this rule in action

Notice we are multiplying both sides by negative two. Notice how the greater than > symbol changed to a less than symbol <. Keep in mind that the variable can be on either side. X<6 is the same as 6>X. This is read X is less than 6, so generally we want the X on the left side, so be careful to keep the inequality facing the correct way if you change it.

Try some questions...

Use paper and pencil if needed.

Multiple Choice

Which inequality is the same as X > 4

• 4 < X

• X ≥ 4

• 4 > X

• X = 4

Multiple Choice

Solve for X

3x+4<10

• X > 2

• X < 2

• X < - 2

• X > - 2

Multiple Choice

Solve for X

- 4X ≤ 12

• X ≤ 3

• X ≥ 3

• X ≤ - 3

• X ≥ - 3

Multiple Choice

When you multiply or divide both sides by a negative number

in an inequality, you should

• Flip the fraction

• Change the direction of the inequality

• Eat a big breakfast

• Crocodiles have V shaped noses

Graphing inequalities on a number line.

Notice the X>3 and X<3 have an open circle, that is because 3 is not included in the inequality. Notice that the X≥3 and X≤3 have a closed circle, that is because 3 is included in the inequality. Make sure you use the correct circle based on the inequality symbol. By placing the circle above the number line, it will be easier to tell if it is open or closed.

What does X > 3 actually mean?

Notice that our arrows on our number lines start at 3 and then go left or right depending on the inequality sign. This means that X could be any number the arrow covers, unlike equal signs which means X can only equal one number. X can not equal a number the arrow does not cover. Pro-tip: if X is on the left side of the inequality, then the arrow matches the direction of the inequality.

combine what we learned

First solve the inequality for the given variable then graph it on a number line. Just remember to change inequality directions when multiplying or dividing by a negative and use open circles for < and > or closed circles for ≤ and ≥.

Try some questions...

Use pencil and paper if needed.

Multiple Choice

< and > use what kind of circles?

• closed

• filled

• open

• colored

Multiple Choice

what is the correct inequality for this graph?

• x > -4

• x < -4

• x ≤ -4

• x ≥ -4

Multiple Choice

Which graph represents the inequality

X > 2

Multiple Choice

Which picture best shows the answer to this inequality?

-2x ≤ - 4