6.5 – Solving Equations w/ Rational Expressions LCD: 20

LCD: 6.5 – Solving Equations w/ Rational Expressions

LCD: 6x 6.5 – Solving Equations w/ Rational Expressions

LCD: x – Solving Equations w/ Rational Expressions

LCD: 6.5 – Solving Equations w/ Rational Expressions

LCD: abxSolve for a 6.5 – Solving Equations w/ Rational Expressions

Problems about Numbers If 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 Solving

Problems 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 2 3 x 6.6 – Rational Equations and Problem Solving

Problems about Work Time to sort one batch (hours) Fraction of the job completed in one hour Ryan Mike Together 2 3 x hrs. LCD =6x 6.6 – Rational Equations and Problem Solving

James 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 hour James Andy Together 2 8 x 6.6 – Rational Equations and Problem Solving

Time to mow one acre (hours) Fraction of the job completed in one hour James Andy Together 2 8 x LCD: hrs. 8x 6.6 – Rational Equations and Problem Solving

A 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 hour 1 st pump 2 nd pump Together x – Rational Equations and Problem Solving

Time to pump one basement (hours) Fraction of the job completed in one hour 1 st pump 2 nd pump Together x – Rational Equations and Problem Solving

LCD: hrs. 60x 6.6 – Rational Equations and Problem Solving

Distance, 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 Solving

A car travels six hundred miles in the same time a motorcycle travels four hundred and fifty miles. If the car’s speed is fifteen miles per hour faster than the motorcycle’s, find the speed of both vehicles. RateTimeDistance Motor- cycle Car x x mi 600 mi t t 6.6 – Rational Equations and Problem Solving

RateTimeDistance Motor- cycle Car x x mi 600 mi t t LCD:x(x + 15) 6.6 – Rational Equations and Problem Solving

x(x + 15) Motorcycle Car 6.6 – Rational Equations and Problem Solving

RateTimeDistance Up Stream Down Stream A 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 = x x - 5 x mi 42 mi t t 6.6 – Rational Equations and Problem Solving

RateTimeDistance Up Stream Down Stream boat speed = x x - 5 x mi 42 mi t t LCD:(x – 5)(x + 5) 6.6 – Rational Equations and Problem Solving

Boat Speed (x – 5)(x + 5) 6.6 – Rational Equations and Problem Solving

Direct Variation: y varies directly as x (y is directly proportional to x), if there is a nonzero constant k such that 6.7 – Variation and Problem Solving The number k is called the constant of variation or the constant of proportionality

Direct Variation 6.7 – Variation and Problem Solving Suppose 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 equation constant of variation x y

6.7 – Variation and Problem Solving Hooke’s 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 equation constant of variation

Inverse Variation: y varies inversely as x (y is inversely proportional to x), if there is a nonzero constant k such that 6.7 – Variation and Problem Solving The number k is called the constant of variation or the constant of proportionality.

Inverse Variation 6.7 – Variation and Problem Solving Suppose 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 equation constant of variation x y

6.7 – Variation and Problem Solving The 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 equation constant of variation

Joint Variation 6.7 – Variation and Problem Solving If 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 Variation 6.7 – Variation and Problem Solving The 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 cubic inches. What is the volume of a can that has a 2 inch radius and a 10 inch height?

Additional Problems

LCD: – Solving Equations w/ Rational Expressions

LCD: x 6.5 – Solving Equations w/ Rational Expressions

LCD: Not a solution as equations is undefined at x = – Solving Equations w/ Rational Expressions

Problems about Numbers The quotient of a number and 2 minus 1/3 is the quotient of a number and 6. Find the number. LCD = – Rational Equations and Problem Solving

7.1 – Radicals Radical 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 index radical sign radicand The 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 – Radicals

Square 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 – Radicals

Rdicals Cube Roots A cube root of any positive number is positive. Examples: A cube root of any negative number is negative. 7.1 – Radicals

n th Roots An n th root of any number a is a number whose n th power is a. Examples: 7.1 – Radicals

n th Roots An n th root of any number a is a number whose n th power is a. Examples: Non-real number 7.1 – Radicals

6.4 – Synthetic Division

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