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Chapter 1: Linear Functions, Equations, and Inequalities

Chapter 1: Linear Functions, Equations, and Inequalities. 1.1 Real Numbers and the Rectangular Coordinate System 1.2 Introduction to Relations and Functions 1.3 Linear Functions 1.4 Equations of Lines and Linear Models 1.5 Linear Equations and Inequalities

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Chapter 1: Linear Functions, Equations, and Inequalities

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  1. Chapter 1: Linear Functions, Equations, and Inequalities 1.1 Real Numbers and the Rectangular Coordinate System 1.2 Introduction to Relations and Functions 1.3 Linear Functions 1.4 Equations of Lines and Linear Models 1.5 Linear Equations and Inequalities 1.6 Applications of Linear Functions

  2. 1.6 Applications of Linear Functions Solving Application Problems • Read the problem and make sure you understand it. Assign a variable to what you are being asked to find. If necessary, write other quantities in terms of this variable. • Write an equation that relates the quantities described in the problem. You may need to sketch a diagram and refer to known formulas. • Solve the equation and determine the solution. • Look back and check your solution. Does it seem reasonable?

  3. 1.6 Dimensions of a Television Screen • The new generation of televisions has a 16:9 aspect ratio. The length of its rectangular screen is times its width. If the perimeter of the screen is 136 inches, find the length and width of the screen. Analytic Solution

  4. 1.6 Dimensions of a Television Screen Graphical Solution Notice that the point of intersection is the point (24.48,136). The x-coordinate supports our previous result from the analytic solution. 200 50 0

  5. 1.6 A Mixture-of-Concentrations Problem • How much pure alcohol should be added to 20 liters of 40% alcohol to increase the concentration to 50% alcohol? • Let x represent the number of liters of pure alcohol to be added Liters of Liquid 20 .40 x 1.0 20 + x .50 Alcohol Concentration

  6. 1.6 Break-Even Analysis • Peripheral Visions, Inc., produces high-definition DVDs of live concerts. The company places an ad in a trade newsletter. The cost of the ad is $100. Each DVD costs $20 to produce, and the company charges $24 per DVD. • Express the cost C as a function of x, the number of DVDs produced. • Express the revenue R as a function of x, the number of DVDs sold. • For what value of x does revenue equal cost?

  7. 1.6 Break-Even Analysis • Graph in an appropriate window to support your answer. • Use a table to support your answer. 1200 0 95 0

  8. 1.6 Direct Variation A number yvaries directly with x if there is a nonzero number k such that The number k is called the constant of variation

  9. 1.6 Direct Variation Example • Hooke’s Law states that the distance (y) a spring stretches varies directly with the force (x) applied. If a force of 15 lbs stretches a spring 8 inches, how much will a force of 35 lbs stretch the spring?

  10. 1.6 Using Similar Triangles • A grain bin in the shape of an inverted cone has height 11 feet and radius 3.5 feet. If the grain is 7 feet high in the bin, calculate the volume of the grain. 3.5 ft. 11 ft. 11 ft.

  11. 1.6 Solving a Formula for a Specified Variable • A trapezoid has area 169 square inches, height 13 inches, and base 19 inches. Find the length of the other base by solving the formula for and substituting.

  12. 1.6 Solving a Formula for a Specified Variable • Solve each formula for the specified variable.

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