Review Homework

page 203-206 # 1-12

1) + = 6000 x 6000+x $

% 14 10 12 6000(14) + 10x = 12(6000 + x) 84000 +10 x = 72000 + 12x 12000 = 2x x = 6000 $6000

0

9)

Principal Rate Time Interest

1.5% X 0.015 1 0.015x

3% 22500 – x 0.03 1 0.03(22500-x)

15x = 30(22500) – 30x45x = 675000X = 15000

$15,000 at 1.5%$7500 at 3%

10x + 30( 10 –x) = 15(10)11

10

Solving Inequalities in Solving Inequalities in one variableone variable

Graph linear inequalities in one Graph linear inequalities in one variablevariable

Solve linear inequalities in one Solve linear inequalities in one variablevariable

Page 208

Inequalities

Equations show two expressions have the same value.

Inequalities compare the values of two expressions.

What’s an inequality?

• Is a range of values,

rather than ONE number

• An algebraic relation showing that a quantity is greater than or less than

another quantity.

Speed limit:

7055 x

> is greater than, is more than, must exceed

< is less than, must be less than

≥ is greater than or equal to, at least, minimal, smallest

≤ is less than or equal to, cannot exceed, maximum, at most

≠ is not equal to

graphing on a number line

If you have <, >, or ≠ then you place on the number line at the number from the problem If you have ≤, ≥, or = then you place on the number line at the number from the problem

You then draw a ray or rays(≠) in the directions of the solution

Examples

x > 1

x<1

x≤-1

x≥-2

x≠3

Note that the “>” can be replaced by , <, or .

Examples: Linear inequalities in one variable.

2x – 2 < 6x – 5

A linear inequality in one variable is an inequality which can be put into the form

ax + b > c

can be written – 4x + (– 2) < – 5.

6x + 1 3(x – 5)

2x + 3 > 4

can be written 3x + 1 – 15.

where a, b, and c are real numbers.

A solution of an inequality in one variable is a number which, when substituted for the variable, results in a true inequality.Examples: Are any of the values of x given below solutions of 2x > 5?

2 is not a solution.

2.6 is a solution.

x = 2 2(2) > 5 4 > 5

x = 2.6 2(2.6) > 5 5.2 > 5

The solution set of an inequality is the set of all solutions.

3 is a solution.x = 3 2(3) > 5 6 > 5

x = 1.5 2(1.5) > 5 3 > 5 1.5 is not a solution.False

False

True

True

? ?

? ?

? ?

? ?

Solutions to Linear Inequalities

write and graph each of the write and graph each of the following inequalities.following inequalities.

The summer temperature T, in The summer temperature T, in Phoenix is greater than or equal to Phoenix is greater than or equal to 80 degrees.80 degrees.

The average snow fall is less than The average snow fall is less than one inch.one inch.

The class average is greater than or The class average is greater than or equal to 85%.equal to 85%.

T ≥ 80 degrees 77

78 79 80

81 82

S < 1 inch -2 -1 0 1 2 3 4

A ≥ 85% 82

83 84 85

86 87

Write and Graph a Linear Inequality

Sue ran a 2 km race in 8 minutes. Write an inequality to describe the average speeds of runners who were faster than Sue. Graph the inequality.

Faster average speed >

Distance

Sue’s Time 8

2s

4

1s

-5 -4 -3 -2 -1 0 1 2 3 4 5

Solving Inequalities

Use the same steps as solving an equation EXCEPT if you multiply or divide by a negative number the direction of the inequality sign changes.

-2x > 4 x< -2Why do we switch the direction of the inequality sign??

2-4x > 6-2 -2-4x > 4 (Multiply by – ¼ )x < -1

2-4x > 6 +4x +4x2> 6 + 4x-6 -6-4 > 4x-4 > 4x (Multiply by ¼ )-1 > x x<-1

Multiplying a quantity by -1 changes it into its opposite. For example, 3 becomes the opposite of 3 or -3, and 12 becomes the opposite of 12, which is -12. Now think about how these numbers fall on a number line:----------------------0----1----2----3--------------------12-----

Since 3 < 12, three is closer to zero and twelve is farther away. That's true in general. Numbers with larger magnitudes are farther away from zero.

Now what happens if we take opposites? That is, if we multiply or divide these numbers by -1? Opposites are the same distance from zero so -12 will be farther away from 0 than -3: --- -12 --------------- -3 ----- 0 ---------------------------

This is true in general. If A < B and both are positive, then the opposite of B will be farther to the left than the opposite of A, that is, -B < -A which means the same thing as -A > -B. If you just look at the symbols it seems as if we put minus signs in front of each letter and switch the inequality symbol. But if you look at the meaning of the symbols on the number line it is much clearer: -B -A 0 A B ---------------------------------------------------------

• Page 210 Two examples show how to solve and graph.

• 3 problems at the bottom – solve and graph on a number line.

Verbal Inequalities Page 211

• Yesung and Gunho decide to order a pizza. They have ₩12,000 to spend. A large cheese pizza costs ₩8000 plus ₩600 for each additional topping. Write an inequality to represent this situation and solve it to determine the maximum number of toppings they can order.

solution

Homework

•Page 212-213