Transcript Solving Inequalities

Solving Inequalities
Solving Inequalities Examples




Joe and Katie are dancers. Suppose you compare
their weights.
Let j stand for Joe’s weight and k stand for Katie’s
weight.
Then you can use inequalities and an equation to
compare their weights.
You can make only one of the following statements.





Joe’s weight is less than Kate’s weight (<).
Joe’s weight is less than or equal to Kate’s weight (≤).
Joe’s weight is the same as Kate’s weight (=).
Joe’s weight is greater than Kate’s weight (>).
Joe’s weight is greater than or equal to Kate’s
weight (≥).
Addition and Subtraction
Properties for Inequalities





If a > b, then a + c > b + c and
a – c > b – c.
If a ≥ b, then a + c ≥ b + c and
a – c ≥ b – c.
If a < b, then a + c < b + c and
a – c < b – c.
If a ≤ b, then a + c ≤ b + c and
a – c ≤ b – c.
Adding or subtracting the same number to each side
of an inequality does not change the truth of the
inequality.
Numerical Examples to Illustrate the Addition
and Subtraction Properties
Addition
 5<6
Subtraction
 –6 > –10

5+2<6+2

–6 – 9 > –10 – 9

7<8

–15 > –19
These properties can be used to solve
inequalities. Each solution set can be graphed on
the number line.
Inequality Graphing Symbols
Verbal
Phrase
Less Than
Inequality Graphing
Symbol
Symbol
<
Meaning
Does not
include
Less Than or
Equal To
≤
Does
include
Greater Than
>
Does not
include
Greater Than
or Equal To
≥
Does
include
Example to Graph a Solution Set
Solve y – 6 > 3. Graph the solution set.
y-6>3
y–6+6>3+6
y >9
Boundary
Add 6 to both sides.
When variable is on left of inequality
graph follows inequality symbol.
Point
6
7
8
9
10
11
12
Solve the Following Inequality and Graph
b + 6 > 11
b + 6 – 6 > 11 – 6 Add -6 to both sides.
b>5
With variable on left of
inequality the symbol points in
the direction of your graph.
1
2
3
4
5
6
7
8
9
Check Inequalities

To check inequalities, first check the boundary point
for the variable, and see if the two sides are
equivalent. A true equation should occur if the
inequality sign is replaced by the equals sign. To
make sure the direction of the inequality is correct,
check a point on each side of the boundary point.
Check
9-6=3
3=3
True
Choose a point in
the solution set.
10 - 6 > 3
4>3
TRUE
Choose a point
outside the
solution set.
7–6>3
1>3
FALSE
Example to Graph a Solution Set
Solve 9x + 7 < 8x – 2. Graph the solution set.
9x + 7 < 8x – 2
–8x + 9x + 7 < –8x + 8x – 2
Add -8x to both sides.
x + 7 < –2
x + 7 + (–7) < –2 + (–7)
Add -7 to both sides.
x < –9
When variable is on left graph
follows inequality symbol.
-13 -11 -10 -9 -8 -7 -6 -5 -4
Check Solution 9x + 7 < 8x - 2
Check Boundary
9(–9) + 7 = 8(–9) – 2
–74 = –74
True
Choose a point
outside the
solution set.
9(–8) + 7 < 8(–8) - 2
–65 < –66
Choose a point in the
FALSE
solution set.
9(–10) + 7 < 8(–10) - 2
-90 + 7 < -80 - 2
–87 < -82
TRUE
Solve the Following Inequality,
Graph and Check Solutions
Solve 7x + 8 ≥ 16 + 6x. Graph and check the solution.
7x + 8 ≥ 16 + 6x
7x + 8 – 6x ≥ 16 + 6x – 6x Add -6x to both sides.
x + 8 ≥ 16
x + 8 – 8 ≥ 16 – 8
Add -8 to both sides.
x≥8
1
2
3
4
5
6
7
8
9 10 11 12 13 14
Check Solution 7x + 8 ≥ 16 + 6x
Check Boundary
7(8) + 8 = 16 + 6(8)
56 + 8 = 16 + 48
64 = 64
True
Choose a point in the
solution set.
7(9) + 8 ≥ 16 + 6(9)
63 + 8 ≥ 16 + 54
71 ≥ 70
TRUE
Choose a point outside
the solution set.
7(5) + 8 ≥ 16 + 6(5)
35 + 8 ≥ 16 + 30
43 ≥ 46
FALSE
Multiplication Properties for
Inequalities
 If
c is positive and a < b, then
ac < bc and ca < cb.
 If c is positive and a > b, then
ac > bc and ca > cb.
 If c is negative and a < b, then
ac > bc and ca > cb.
 If c is negative and a > b, then
ac > bc and ca < cb.
Numerical Example of
Multiplication Property
We know that 18 > –11 is a true inequality.
 If you multiply each side of this inequality by a
positive number, the result is a true inequality.

18 > -11
18 (3) > -11 (3) Multiply both sides by 3.
54 > -33
True Inequality
Solve the Inequality
Solve x / 3 > –27. Graph the solution set.
x / 3 > - 27
3 (x / 3) > -27 (3) Multiply both sides by 3.
x > - 81
-85 -84 -83 -82 -81
-80 -79 -78
-77
Check Solution x / 3 > -27
Check Boundary
-81 / 3 = -27
-27 = -27
True
Choose a point
in the solution
-78 / 3 > -29
-28 > -29
True
Choose a point
outside solution
-84 / 3 > -27
-28 > -27
False
What Happens if We Multiply by a
Negative Number?

Suppose you multiply each side of a true
inequality by a negative number.
18 > –11
18(–2) > –11(–2)
–36 > 22
FALSE!
18 > –11
18(–2) > –11(–2)
–36 < 22
True
We must reverse the inequality symbol when we
multiply by a negative number.