NOTES 1-7: Solving & Graphing Inequalities · 2017-12-22 · NOTES 1-7: Solving & Graphing Inequalities What is an inequality? (What is a compound inequality? za Graphing Inequalities:

NOTES 1-7: Solving & Graphing Inequalities

What is an inequality?

What is a compound inequality?

Graphing Inequalities:

Graph the inequality on the given number line.

Ex. 1: 𝑥𝑥 > 5

Ex. 2: 𝑥𝑥 ≤ −3

Ex. 3: 𝑥𝑥 < 2 𝑜𝑜𝑜𝑜 𝑥𝑥 ≥ 6

Ex. 4: 2 ≤ 𝑥𝑥 < 8

Write the inequality for each graph.

Ex. 5:

Ex. 6:

-10 -5 0 5 10 x

-10 -5 0 5 10 x

-10 -5 0 5 10 x

-10 -5 0 5 10 x

-10 -5 0 5 10 x

-10 -5 0 5 10 x

An inequality is like an equation, but we can have

a Lot of solutions .instead ofusing an = sign ,

we use > (greater than ),

2 (greaterthan or equal to ) , < ( less than )

,and I ( Less than or equal to ) .

when we have 2 possibe inequality solutions .

An intersection looks like za × ± 8 , which means X > 2 and XE 8 .

A union says the solution can be in one of two places : example

=×< 20±⇐-

< or >, we putan open circle on the number line

#%

1 or 2,we put a closed circle on the number line

¥- shade where the inequalityis true . ×±z

× > 5

.

Ommmx

xistall numbers biggerthan 5 . Open circle .

TX is all numbers Smaller →

than orequaltt -3 .

Closed circle .

X can be smaller than Dt Q •×z6mmx

or bigger thanorequal-to 6

FO× is between 2 and21 × < 8

8 .

It can also equal 2 .

gassed× > I

OMXEMO

- 5<×< 1

Ex. 7:

Solving Inequalities:

The following properties are used to justify the steps we take when solving

equations AND inequalities:

• Addition • Division • Subtraction • Distributive Property • Multiplication • Combine Like Terms

Solve each inequality, justifying each step, then graph the inequality on a

number line, and check your answer.

−11𝑦𝑦 − 9 > 13 Justification

Ex. 8:

-10 -5 0 5 10 xO.O Dh@¥t5Eesff

X< - 4 or X 22

solve just like equations ,EXCEPT when you multiply

or divide by a negative number, you must switch the

direction of the inequality .

why ? Negativestell us to

go the opposite direction .

+9 +9-

Addition

-

ny > 22 Division (By a negative, soTt change

the sign! )

*07*6 Check it ! we shaded over -3

.

Is

-3<-2 ? yes ! so we shaded the

rightway .

7𝑥𝑥 + 9 ≥ 10𝑥𝑥 − 12 Justification

Solving Compound Inequalities:

6 < 3𝑥𝑥 − 9 ≤ 12 Justification

Ex. 9:

Ex. 10:

- a - 7- Subtraction ( l subtracted the 7× to keepthe × term positive)

923×-12

+12 +12Addition

÷-3X Division ( By a positive number ,

so leave

- J the inequalityalone .

):ZX But this is backwards .

It says that 7 is bigger

[email protected] Over 6.Is 726 ?

¥537387 yes ! we shaded the right way .

Apply operations to ad I parts of the inequality.

Remember : 2ex< 8 means that 2±×a±dX< 8.

we could solve

I

each part separately , but we'd use the same steps to get Xby itself

so we might as well do it all

ato :::+9Tin Addition15L.3×121..

DivisionJ ::X#

Check ! we shaded over 6 . Plug it intothe

first equation .Is 6<3161 -9<-12

tem > 6<18-91126<9112 ?

. Yes !