College Algebra

3.6 Polynomial and Rational Inequalities

3.7 Variation

3.6 Polynomial InequalitiesObj: solve polynomial and rational inequalities with the critical

value method

• Critical Value Method– Zeros (solutions) of polynomials are called the

critical points.

– Critical Points of a polynomial divide positive values from negative values.

Example

Solve x2 – 2x – 15 < 0

• Factor and solve for x. These are the critical points.• Choose a value in each interval.

Let x = • Plug each value into the factored form ( + or - ).• Compare relations.• Write solution in interval notation.

Rational Inequalities

Steps:1. Set = to 0 and simplify.2. Find the zeros of the numerator and denominator to

find the critical points.3. Test the intervals.4. Compare the relations.5. Write the solution.

Example

Solve 35

95

x

x

3.7 VariationObj: to set up and solve problems using variation

• Direct Variation

• Inverse Variation

• Joint Variation

• Combined Variation

Direct Variation

Varies directly or is directly proportional toMeaning as one unit increases, the other

increases or as one decreases, the other decreases also

y = kx k is the constant of

proportionality

Examples:

ExampleThe distance d that a ball rolls down an inclined

plane is directly proportional to the square of time t. If the ball rolls 5 feet in 1 second, how far will it roll in 4 seconds?

Set up the equation using initial info.

Solve for k.

Set up new equation with k value and new info.Solve.

Inverse VariationVaries inversely or is inversely proportional toMeaning as one unit increases, the other

decreases and vice versa.

y =

Examples:

x

k

ExampleThe speed v of a gear varies inversely as the

number of teeth t. If a gear that has 48 teeth makes 20 revolutions per minute, how many revolutions per minute will a gear that has 30 teeth make?

Joint Variationmore than one variable

z = kxy

The cost of a concrete patio varies jointly as the area of the patio and the depth of the patio. It costs $500 for a patio with an area of 80 square feet and a depth of 4 inches. Find the cost of a patio with an area of 144 square feet and a depth of 6 inches.

Combined Variationmore than one type of variation

The volume of a given mass of a gas varies directly as the temperature T and inversely as the pressure P. If the volume of the gas is 220 cm3 when T = 40˚C and P = 20 kg/cm2, what is the volume when T = 35˚C and P = 10 kg/cm2?

Assignment

• 3.6 page 420 1 – 13 eoo, 29 – 45 eoo

• 3.7 page 430 1 – 29 eoo