college algebra

COLLEGE ALGEBRA1.1 Linear and Absolute Value Equations1.2 Formulas and Applications1.3 Quadratic Equations1.4 Other Types of Equations

1.1 Linear EquationsAn equation is a statement about the equality of two expressions.

Within each equation we have true and false values. The value which satisfies the equation gives a true value. For all other values, give false values.

To solve an equation means to find all values of the variable that satisfy the equation. The values that satisfy an equation are called solutions or roots of the equation.

Equivalent equations are equations that have exactly the same solution or solutions. Our goal is to have this

Variable = Constant

1.1 Linear EquationsA Linear Equation, or first-degree equation, in the single variable x is an equation that can be written in the form.

where a and b are real numbers, with

1.1 Linear EquationsSolve:

1.1 Linear EquationsSolve:Hint…use the distributive property to solve…

1.1 Linear EquationsSolve by clearing fractions:

1.1 Linear EquationsSolve by clearing fractions:

1.1 Contradictions, Conditional Equations, and Identities

There are 3 types of equations:An equation that has no solutions is called a contradiction.

An equation that is true for some values of the variable but not true for other values of the variable is called a conditional equation.

An identity is an equation that is true for all values of the variable for which all terms of the equation are defined.

1.1 Contradictions, Conditional Equations, and Identities

Classify the equation:

1.1 Contradictions, Conditional Equations, and Identities

Classify the equation:

1.1 Contradictions, Conditional Equations, and Identities

Classify the equation:

1.1 Absolute Value EquationsRemember the absolute value of a real number x is the distance between the number x and the number 0 on the real number line.

Property:For any variable expression E and any nonnegative real number k,

if and only if

1.1 Absolute Value EquationsSolve:

1.1 ApplicationMovie theatre ticket prices have been increasing steadily in recent years. An equation that models the average U.S. movie theater ticket prices p, in dollars is given by:

where t is the number of years after 2003. (This means that t = 0 corresponds to 2003.) Use this equation to predict the year in which the average U.S. movie theater ticket prices will reach $7.50

1.1 ApplicationAlicia is driving along a highway that passes through Centerville. Her distance d, in miles, from Centerville is given by the equation:

where t is the time in hours since the start of her trip and Determine when Alicia will be exactly 15 miles from Centerville.

1.2 FormulasA formula is an equation the expresses known relationships between two or more variables.

Page 84 of your textbook has a listed of these formulas.

1.2 FormulasSolve

1.2 FormulasSolve

1.2 FormulasThe NFL uses the following formula to rate quarterbacks.

C is the percentage of pass completions, Y is the average number of yards gained per pass attempt, T is the percentage of touchdown passes, and I is the percentage of interceptions.During the 2008 season, Philip Rivers, the quarter back of the San Diego Chargers, completed 65.3% of his passes. He averaged 8.39 yards per pass attempt, 7.1% of his passes were for touchdowns, and 2.3% of his passes were intercepted. Determine Rivers’s quarterback rating for the 2008 season.

1.2 ApplicationsStrategies for Solving Applications:1. Read the problem carefully.2. When appropriate, draw a sketch and label parts of the

drawing with specific information given in the problem.3. Determine the unknown quantities, and label them with

variables. Write down any equation that relates the variables.

4. Use the information from Step 3, along with a known formula or some additional information given in the problem, to write an equation.

5. Solve the equation obtained in Step 4, and check to see whether the results satisfy all the conditions of the original problem.

1.2 ApplicationsOne of the best known paintings is the Mona Lisa by Leonardo da Vinci. It is on display at the Musee du Louvre, in Paris. The length (or height) of this rectangular-shaped painting is 24 centimeters more than its width. The perimeter of the painting is 260 centimeters. Find the width and length of the painting.

1.2 Similar TrianglesSimilar triangles are one for which the measures of corresponding angles are equal. The triangles below are similar.

Ratio Relationships

ad e

fc

b

1.2 Similar TrianglesA person 6 feet tall is in the shadow of a building 40 feet tall and is walking directly away from the building. When the person is 30 feet from the building, the tip of the person’s shadow is at the same point as the tip of the shadow of the building. How much farther must the person walk to be just out of the shadow of the building? Round to the nearest tenth of a foot.

1.2 Similar Triangles

1.2 ApplicationsIt costs a tennis show manufacturer $26.55 to produce a pair of tennis shoes that sells for $49.95. how many pairs of tennis shoes must the manufacturer sell to make a profit of $14,274.00?Profit = Revenue – Cost

1.2 ApplicationsSimple interest formula:

Where I is the interest, P is the principle, r is the simple interest rate per period, and t is the number of periods.

An accountant invests part of a $6000 bonus in a 5% simple interest account and invests the remainder of the money at 8.5% simple interest. Together the investments earn $370 per year. Find the amount invested at each rate.

1.2 ApplicationsA runner runs a course at a constant speed of 6 mph. One hour after the runner begins, a cyclist starts on the same course at a constant speed of 15 mph. How long after the runner starts does the cyclist overtake the runner?

Rate r X Time t = Distance dRunner

Cyclist

1.2 ApplicationsA chemist mixes an 11% hydrochloric acid solution with a 6% hydrochloric acid solution. How many milliliters of each solution should the chemist use to make a 600 – milliliter solution that is 8% hydrochloric acid?

1.2 ApplicationsHow many ounces of pure silver costing $10.50 per ounce must be mixed with 60 ounces of a silver alloy that costs $7.35 per ounce to produce a silver alloy that costs $9.00 per ounce?

1.2 ApplicationsWork ProblemsRate of Work X time worked = part of task completed

Pump A can fill a pool in 6 hours, and pump B can fill the same pool in 3 hours. How long will it take to fill the pool if both pumps are used?

1.3 Solving Quadratic Equations by FactoringA quadratic equation in x is an equation that can be written in the standard quadratic form.

where a, b, and c are real numbers and

The Zero Product PrincipleIf A and B are algebraic expressions such that AB = 0, then A = 0 or B = 0.

1.3 Solving Quadratic Equations by FactoringSolve by factoring:

1.3 Solving Quadratic Equations by FactoringSolve by factoring:

1.3 Solving Quadratic Equations by FactoringSolve by factoring:

A double solution or double root happens when there are two identical solutions.

1.3 Solving Quadratic Equations by Taking Square Roots

The Square Root Procedure:If

Example: which can be written as which can be written as which can be written as

1.3 Solving Quadratic Equations by Taking Square Roots

Use the square root procedure to solve the equation.

1.3 Solving Quadratic Equations by Taking Square Roots

Use the square root procedure to solve the equation.

1.3 Solving Quadratic Equations by Completing the Square

Consider the following:

Important that when completing the square we notice the co-efficient of the first term is 1.Adding to a binomial of the form the constant term that makes the binomial a perfect-square trinomial is called completing the square. You can use completing the square to complete ANY quadratic equation.

Square of a Binomial = Perfect Square Trinomial

1.3 Solving Quadratic Equations by Completing the Square

Solve by completing the square:

1.3 Solving Quadratic Equations by Using the Quadratic Formula

The quadratic formula can be used to complete ANY quadratic equation.If then

1.3 Solving Quadratic Equations by Using the Quadratic Formula

Solve by using the Quadratic Formula:

1.3 Solving Quadratic Equations by Using the Quadratic Formula

Solve by using the Quadratic Formula:

1.3 The Discriminant of a Quadratic EquationThe expression under the radical, , is called the discriminant. The discriminant is used to determine whether the solutions of a quadratic equation are real numbers.

If has two distinct real solutions.

If has one real solution. This solution is a double solution.

If has two distinct non real complex solutions. The solutions are conjugates of each other.

1.3 The Discriminant of a Quadratic EquationUse the discriminant to determine the number of real solutions.

1.3 The Discriminant of a Quadratic EquationUse the discriminant to determine the number of real solutions.

1.3 Pythagorean TheoremIf a and b denote the lengths of the legs of a right triangle and c the length of the hypotenuse, then

1.3 Pythagorean TheoremA television screen measures 60 inches diagonally, and its aspect ratio is 16 to 9. This means that the ratio of the width of the screen to the height of the screen is 16 to 0. Find the width and height of the screen.

1.3 Pythagorean TheoremA company makes regular solid candy bars that measure 5 inches by 2 inches by 0.5 inches. Due to difficult financial times, the company has decided to keep the price of the candy bar fixed and reduce the volume of the bar by 20%. What should the dimensions be for the new candy bar if the company keeps the height at 0.5 inch and makes the length of the candy bar 3 inches longer than the width?

1.4 Polynomial EquationsSolve:

1.4 Rational EquationsA rational equation is one that involves rational expressions.

Keep in mind that within the above equation, both -3 and 1 can not be possible values of x since either one would create a division by 0.

1.4 Rational EquationsSolve:

1.4 Rational EquationsSolve:

1.4 Radical EquationsThe Power Principle:If P and Q are algebraic expressions and n is a positive integer, then every solution of P = Q is a solution of Solve:

1.4 Radical EquationsSolve:

1

1.4 Rational Exponent EquationsRemember…. when n is a positive even integer when n is a positive odd integer

Also…, n is a positive even integer, n is a positive odd integer

1.4 Rational Exponent EquationsSolve:

1.4 Rational Exponent EquationsSolve:

1.4 Equations that are Quadratic in FormThe equation is said to be quadratic in form, which means that it can be written in the form:

Where u is an algebraic expression involving x. For example, if we make the substitution (which implies that ), then the original equation can be written as

This quadratic equation can be solved for u and then, using the relationship , we can find the solutions of the original equation.

Our goal when we create a u = is to ensure that the first term exponent is 2.

1.4 Equations that are Quadratic in FormSolve:

1.4 Equations that are Quadratic in FormSolve:

1.4 Other Types of EquationsTwo busses are transporting a football team to a game that is 120 miles away. The second bus travels at an average speed that is 10mph faster than the first bus and arrives 1 hour sooner than the first bus. Find the average speed of each bus.

or

1.1 – 1.4 Homework•Continue finding articles•Chapter 1 Review Exercises

• 1 – 39, ODDS• 57 – 75, ODDS