math. algebra

MATH

1. How many liters of 20% alcohol solution should be added to 40 liters of a 50% alcohol solution to make a 30% solution?

40L50%

x L20%

(40+x) L

30%+ =

x kgP192

(8-x) kg

P168

8kgP174

+ =

3. Poppy can mow the lawn in 40 minutes and Ryann can mow the lawn in 60 minutes. How long will it take for them to mow the lawn together?

4. An airplane which travels 180 miles per hour leaves the airport 8.5 hours after a ship sailed. If it overtakes the ship in 1 hour and 30 minutes, find the rate of the ship.

RATE TIME DISTANCE

Airplane180mi/

h1.5h 1.5(180)

Ship x1.5h + 8.5h

10x

5. A bus traveling at an average rate of 50 kilometers per hour made the trip to town in 6 hours. If it had traveled at 45 kilometers per hour, how many more minutes would it have taken to make the trip?

RATE TIME DISTANCE

A 50kph 6h 300km

B 45kph 6h + x45(6 +

x)

Linear InequalityAn INEQUALITY is a mathematical

statement which states that two quantities are not equal.

TRICHOTOMY AXIOMFor any numbers x and y, one and only one of the following is true:

x < yx > yx = y

Linear InequalityREMEMBER!

< - less than > - greater than ≤ - less than or equal to

≥ - greater than or equal to

≠ - not equal to

When 3 is added to x, the sum remains larger than 9.

Linear InequalityWhen you triple d and

subtract 6, the result remains less than or equal to 15.

Linear Inequality

Either s is greater than 8 or less than -8.

t is greater than -1 but less than 5.

Linear Inequality

REMEMBER!

< or > ---------- ( parentheses )

≤ or ≥ ---------- [ brackets ]

Linear Inequalityx > 4x ≥ 58 < xx ≥ 7

0 ≤ x < 5

Linear Inequality

x < 7

-5 ≥ x

Linear Inequality

-5 < x < 7

x > 9 or x < -9

Linear Inequality

Linear Inequality

Linear InequalityAssignment:

What are the properties of inequalities used in solving inequalities?

Linear Inequality

1. TRANSITIVE PROPERTY OF INEQUALITY (TPI)

2. ADDITION PROPERTY OF INEQUALITY (API)

3. MULTIPLICATION PROPERTY OF INEQUALITY (MPI)

Properties of Inequality

Linear Inequality1. TRANSITIVE PROPERTY OF

INEQUALITY (TPI)

For every x, y, and z ∈ ℝ, if x < y and y < z, then x < z.

example:

If 4 < 9 and 9 < 10, then _______. If 5 > 0 and 0 > -5, then _______.

Linear Inequality2. ADDITION PROPERTY OF

INEQUALITY (API)

For every x, y ∈ ℝ and z, any number, if x < y, then x + z < y + z.

example:

If 4 < 9 (z = 2), then _______________. If 5 > 0 (z = -7), then _______________.

Linear Inequality3. MULTIPLICATION PROPERTY OF

INEQUALITY (MPI)

For every x, y ∈ ℝ, if x < y, then xz < yz if z > 0 or xz > yz if z < 0.

example:

If 4 < 9 (z = 2), then _______________. If 5 > 0 (z = -7), then _______________.

Linear InequalityEXAMPLES:

1.2x + 5 > 3x – 3

2.3(x + 2) + 12 ≥ 24

3.1 ≤ x + 1 < 6

4.2x – 1 > 7 or 2 – 3x ≥ -1

{x | x < 11}

{x | x ≥ 2}

{x | 0 ≤ x < 5}

{x | x > 4} or {x | x ≤ 1}

Linear InequalitySEATWORK:

Solve and graph the solution set of each of

the following inequalities.

Get one whole sheet of paper. :)

Linear Inequality