# 6.5 â€“ Solving Equations w/ Rational Expressions

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6.5 – Solving Equations w/ Rational Expressions. LCD: 20. 6.5 – Solving Equations w/ Rational Expressions. LCD:. 6.5 – Solving Equations w/ Rational Expressions. LCD: 6x. 6.5 – Solving Equations w/ Rational Expressions. LCD: x+3. 6.5 – Solving Equations w/ Rational Expressions. LCD:. - PowerPoint PPT Presentation

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6.5 Solving Equations w/ Rational Expressions

LCD: 20

LCD:

6.5 Solving Equations w/ Rational ExpressionsLCD: 6x

6.5 Solving Equations w/ Rational ExpressionsLCD: x+3

6.5 Solving Equations w/ Rational ExpressionsLCD:

6.5 Solving Equations w/ Rational ExpressionsLCD: abx

Solve for a

6.5 Solving Equations w/ Rational ExpressionsProblems about NumbersIf one more than three times a number is divided by the number, the result is four thirds. Find the number.

LCD = 3x

6.6 Rational Equations and Problem SolvingProblems about Work Mike and Ryan work at a recycling plant. Ryan can sort a batch of recyclables in 2 hours and Mike can sort a batch in 3 hours. If they work together, how fast can they sort one batch?

Time to sort one batch (hours)

Fraction of the job completed in one hour

Ryan

Mike

Together

23x6.6 Rational Equations and Problem SolvingProblems about Work

Time to sort one batch (hours)

Fraction of the job completed in one hour

Ryan

Mike

Together

23x

hrs.LCD =

6x

6.6 Rational Equations and Problem SolvingJames and Andy mow lawns. It takes James 2 hours to mow an acre while it takes Andy 8 hours. How long will it take them to mow one acre if they work together?

Time to mow one acre (hours)Fraction of the job completed in one hourJames

Andy

Together

28x

6.6 Rational Equations and Problem Solving

Time to mow one acre (hours)Fraction of the job completed in one hourJames

Andy

Together

28x

LCD:

hrs.

8x

6.6 Rational Equations and Problem SolvingA sump pump can pump water out of a basement in twelve hours. If a second pump is added, the job would only take six and two-thirds hours. How long would it take the second pump to do the job alone?

Time to pump one basement (hours)Fraction of the job completed in one hour1st pump2nd pumpTogetherx12

6.6 Rational Equations and Problem Solving

Time to pump one basement (hours)Fraction of the job completed in one hour1st pump2nd pumpTogetherx12

6.6 Rational Equations and Problem SolvingLCD:

hrs.

60x

6.6 Rational Equations and Problem SolvingDistance, Rate and Time Problems If you drive at a constant speed of 65 miles per hour and you travel for 2 hours, how far did you drive?

6.6 Rational Equations and Problem SolvingA car travels six hundred miles in the same time a motorcycle travels four hundred and fifty miles. If the cars speed is fifteen miles per hour faster than the motorcycles, find the speed of both vehicles. RateTimeDistanceMotor-cycleCarxx + 15450 mi600 mitt

6.6 Rational Equations and Problem SolvingRateTimeDistanceMotor-cycleCarxx + 15450 mi600 mitt

LCD:x(x + 15)

x(x + 15)x(x + 15)6.6 Rational Equations and Problem Solving

x(x + 15)x(x + 15)

Motorcycle

Car6.6 Rational Equations and Problem SolvingRateTimeDistanceUpStreamDownStreamA boat can travel twenty-two miles upstream in the same amount of time it can travel forty-two miles downstream. The speed of the current is five miles per hour. What is the speed of the boat in still water? boat speed = xx - 5x + 522 mi42 mitt

6.6 Rational Equations and Problem SolvingRateTimeDistanceUpStreamDownStreamboat speed = xx - 5x + 522 mi42 mitt

LCD:(x 5)(x + 5)

(x 5)(x + 5)(x 5)(x + 5)6.6 Rational Equations and Problem Solving

Boat Speed

(x 5)(x + 5)(x 5)(x + 5)

6.6 Rational Equations and Problem SolvingDirect Variation: y varies directly as x (y is directly proportional to x), if there is a nonzero constant k such that6.7 Variation and Problem SolvingThe number k is called the constant of variation or the constant of proportionality

Direct Variation

6.7 Variation and Problem SolvingSuppose y varies directly as x. If y is 24 when x is 8, find the constant of variation (k) and the direct variation equation.

direct variation equationconstant of variationxy395159271339

6.7 Variation and Problem SolvingHookes law states that the distance a spring stretches is directly proportional to the weight attached to the spring. If a 56-pound weight stretches a spring 7 inches, find the distance that an 85-pound weight stretches the spring. Round to tenths.

direct variation equationconstant of variation

Inverse Variation: y varies inversely as x (y is inversely proportional to x), if there is a nonzero constant k such that6.7 Variation and Problem SolvingThe number k is called the constant of variation or the constant of proportionality.

Inverse Variation

6.7 Variation and Problem SolvingSuppose y varies inversely as x. If y is 6 when x is 3, find the constant of variation (k) and the inverse variation equation.

direct variation equationconstant of variationxy3692101.8181

6.7 Variation and Problem SolvingThe speed r at which one needs to drive in order to travel a constant distance is inversely proportional to the time t. A fixed distance can be driven in 4 hours at a rate of 30 mph. Find the rate needed to drive the same distance in 5 hours.

direct variation equationconstant of variation

Joint Variation6.7 Variation and Problem SolvingIf the ratio of a variable y to the product of two or more variables is constant, then y varies jointly as, or is jointly proportional, to the other variables.z varies jointly as x and y. x = 3 and y = 2 when z = 12. Find z when x = 4 and y = 5.Joint Variation6.7 Variation and Problem SolvingThe volume of a can varies jointly as the height of the can and the square of its radius. A can with an 8 inch height and 4 inch radius has a volume of 402.12 cubic inches. What is the volume of a can that has a 2 inch radius and a 10 inch height?Additional ProblemsLCD: 15

6.5 Solving Equations w/ Rational ExpressionsLCD: x

6.5 Solving Equations w/ Rational ExpressionsLCD:

Not a solution as equations is undefined at x = 1.6.5 Solving Equations w/ Rational ExpressionsProblems about Numbers The quotient of a number and 2 minus 1/3 is the quotient of a number and 6. Find the number.

LCD = 6

6.6 Rational Equations and Problem Solving7.1 RadicalsRadical Expressions Finding a root of a number is the inverse operation of raising a number to a power.

This symbol is the radical or the radical sign

indexradical signradicandThe expression under the radical sign is the radicand.The index defines the root to be taken.Radical Expressions The symbol represents the negative root of a number.The above symbol represents the positive or principal root of a number.

7.1 RadicalsSquare Roots If a is a positive number, then

is the positive square root of a and

is the negative square root of a.A square root of any positive number has two roots one is positive and the other is negative.Examples:

non-real #

7.1 RadicalsRdicalsCube Roots

A cube root of any positive number is positive.Examples:

A cube root of any negative number is negative.

7.1 Radicalsnth Roots An nth root of any number a is a number whose nth power is a.Examples:

7.1 Radicalsnth Roots

An nth root of any number a is a number whose nth power is a.Examples:

Non-real number

Non-real number

7.1 Radicals

6.4 Synthetic Division6.4 Synthetic Division

Synthetic division will only work with linear factors with an one as the x-coefficient.

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