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Solving Inequalities

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An inequality is like an equation, but instead of an equal sign (=) it has one of these signs: ... Use the addition property of inequality to collect all ... – PowerPoint PPT presentation

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Title: Solving Inequalities

1
Solving Inequalities

Using Addition Subtraction

2
An inequality is like an equation, but instead of
an equal sign () it has one of these signs lt
less than less than or equal to gt
greater than greater than or equal to
3
x lt 5

means that whatever value x has, it must be less
than 5.

4
Numbers less than 5 are to the left of 5 on the
number line.
5
x -2

means that whatever value x has, it must be
greater than or equal to -2.

6
Numbers greater than -2 are to the right of -2 on
the number line.
7
Where is -1.5 on the number line? Is it greater
or less than -2?
-2
8
Solve an Inequality
w 5 lt 8
We will use the same steps that we did with
equations, if a number is added to the variable,
we add the opposite sign to both sides
w 5 (-5) lt 8 (-5)
w 0 lt 3
All numbers less than 3 are solutions to this
problem!
w lt 3
9
More Examples
8 r -2
8 r (-8) -2 (-8)
r 0 -10
r -10
All numbers from -10 and up (including -10) make
this problem true!
10
More Examples
4 y 1
4 y (-4) 1 (-4)
y 0 -3
y -3
All numbers from -3 down (including -3) make this
problem true!
11
Interval Notation
12

A telecommunications company charges 15 monthly
fee plus 0.08 per minute for long distance
calls. A students budget leaves only 35 per
month to spend on long distance. Write an
inequality to describe this situation.

A telecommunications company charges 15 monthly
fee plus 0.08 per minute for long distance
calls. A students budget leaves only 35 per
month to spend on long distance. Write an
inequality to describe this situation.

Let x number of minutes called 15
0.08x 35
14

A telecommunications company charges 15 monthly
fee plus 0.08 per minute for long distance
calls. A students budget leaves only 35 per
month to spend on long distance. Write an
inequality to describe this situation.

Let x number of minutes called 15
0.08x 35 0.08x 20 subtract
15 from both sides
15

A telecommunications company charges 15 monthly
fee plus 0.08 per minute for long distance
calls. A students budget leaves only 35 per
month to spend on long distance. Write an
inequality to describe this situation.

Let x number of minutes called 15
0.08x 35 0.08x 20 subtract
15 from both sides x 20/0.08 250
divide both sides by 0.08
16

A telecommunications company charges 15 monthly
fee plus 0.08 per minute for long distance
calls. A students budget leaves only 35 per
month to spend on long distance. Write an
inequality to describe this situation.

Let x number of minutes called 15
0.08x 35 0.08x 20 subtract
15 from both sides x 20/0.08 250
divide both sides by 0.08 The student must use
less than 250 minutes.
17

We simplified 15 0.08x 35 to x 250.
So for the student to spend 35 or less, they
should use less than or equal to 250 minutes.
Note the student is not forced to use 250 minutes
(250),
just 250 minutes or less (250).

18
Dividing by a negative number

5 lt 7

19
Dividing by a negative number

5 lt 7 is true

20
Dividing by a negative number

5 lt 7 is true
-5 lt -7

21
Dividing by a negative number

5 lt 7 is true
-5 lt -7 dividing both sides by -1 turns the
statement
false

22
Dividing by a negative number

5 lt 7 is true
-5 lt -7 dividing both sides by -1 turns the
statement
false
-5 gt -7 we must reverse the direction of the
inequality

23
Dividing by a negative number

5 lt 7 is true
-5 lt -7 dividing both sides by -1 turns the
statement
false
-5 gt -7 we must reverse the direction of the
inequality
When multiplying or dividing both sides of an
inequality by a negative number, the direction of
the inequality is reversed.

24
Addition Property of Inequalities

If a lt b, then a c lt b c and a c lt b c

25
Addition Property of Inequalities

If a lt b, then a c lt b c and a c lt b c
Example
2x 3 lt 7
2x 3 3 lt 7 3 subtract 3 from both sides
2x lt 4 simplify

26
Addition Property of Inequalities

If a lt b, then a c lt b c and a c lt b c
Example
2x 3 lt 7
2x 3 3 lt 7 3 subtract 3 from both sides
2x lt 4 simplify
In other words, we can add or subtract as long as
we do the same to both sides of the equation.

27
Positive Multiplication Property of Inequalities

If a lt b and c is positive then ac lt bc
and a/c lt b/c
Example 2x lt 4
x lt 2 divide both sides by 2
In other words we can multiply and divide
inequalities by positive values and retain
equivalency.

28
Negative Multiplication Property of Inequalities

If a lt b and c is negative then ac gt bc
and a/c gt b/c

29
Negative Multiplication Property of Inequalities

If a lt b and c is negative then ac gt bc
and a/c gt b/c
Example -4x lt 20
x gt -5 divide both sides by
-4

30
Negative Multiplication Property of Inequalities

If a lt b and c is negative then ac gt bc
and a/c gt b/c
Example -4x lt 20
x gt -5 divide both sides by
-4
In other words we can multiply and divide
inequalities by negative values and but must
switch the direction of the inequality to retain
equivalency.

31
Solving a Linear Inequality

Simplify the algebraic expression on each side.

32
Solving a Linear Inequality

Simplify the algebraic expression on each side.
Use the addition property of inequality to
collect all variable terms on one side and all
constant terms on the other.

33
Solving a Linear Inequality

Simplify the algebraic expression on each side.
Use the addition property of inequality to
collect all variable terms on one side and all
constant terms on the other.
Use the multiplication properties to isolate the
variable and solve.

34
Solving a Linear Inequality

Simplify the algebraic expression on each side.
Use the addition property of inequality to
collect all variable terms on one side and all
constant terms on the other.
Use the multiplication properties to isolate the
variable and solve.
Express the solution set in set builder notation
or interval notation and graph the solution set.

35
Solve and graph 3x 5 gt -17
36
Solve and graph 3x 5 gt -17