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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 !