College algebra

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# College algebra - PowerPoint PPT Presentation

College algebra . 1.1 Linear and Absolute Value Equations 1.2 Formulas and Applications 1.3 Quadratic Equations 1.4 Other Types of Equations. 1.1 Linear Equations. An equation is a statement about the equality of two expressions.

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### College algebra

1.1 Linear and Absolute Value Equations

1.2 Formulas and Applications

1.4 Other Types of Equations

1.1 Linear Equations

An 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 Equations

A 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 Equations

Solve:

Hint…use the distributive property to solve…

1.1 Linear Equations

Solve by clearing fractions:

1.1 Linear Equations

Solve 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 Absolute Value Equations

Remember 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 Application

Movie 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 Application

Alicia 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 Formulas

A 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 Formulas

The 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 Applications

Strategies for Solving Applications:

• When appropriate, draw a sketch and label parts of the drawing with specific information given in the problem.
• Determine the unknown quantities, and label them with variables. Write down any equation that relates the variables.
• Use the information from Step 3, along with a known formula or some additional information given in the problem, to write an equation.
• Solve the equation obtained in Step 4, and check to see whether the results satisfy all the conditions of the original problem.
1.2 Applications

One 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 Triangles

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

Ratio Relationships

e

d

a

b

c

f

1.2 Similar Triangles

A 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 Applications

It 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 Applications

Simple 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 Applications

A 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?

1.2 Applications

A 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 Applications

How 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 Applications

Work Problems

Rate 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 Factoring

A 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 Principle

If A and B are algebraic expressions such that AB = 0, then A = 0 or B = 0.

1.3 Solving Quadratic Equations by Factoring

Solve 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 writte

n 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.

If then

Solve by using the Quadratic Formula:

Solve by using the Quadratic Formula:

1.3 The Discriminant of a Quadratic Equation

The 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 Equation

Use the discriminant to determine the number of real solutions.

1.3 The Discriminant of a Quadratic Equation

Use the discriminant to determine the number of real solutions.

1.3 Pythagorean Theorem

If a and b denote the lengths of the legs of a right triangle and c the length of the hypotenuse, then

1.3 Pythagorean Theorem

A 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 Theorem

A 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 Rational Equations

A 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.

The 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 Rational Exponent Equations

Remember….

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 Equations that are Quadratic in Form

The 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 Other Types of Equations

Two 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

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