Linear Equations and Inequalities

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1 Linear Equations and Inequalities Section 1.1 Prof. Wodarz Math Fall 2008 Contents 1 Linear Equations Standard Form of a Linear Equation Solving Linear Equations Linear Inequalities Linear Inequalities Interval Notation Equivalent Inequalities Solving Linear Inequalities Applications Solving Word Problems Break-Even Analysis Summary 7 1

2 1 Linear Equations 1.1 Standard Form of a Linear Equation Linear Equations Standard Form A first-degree equation or linear equation is one that can be written in the form ax + b = 0 (a 0) This is the standard form of the linear equation. The equation 5 + 3(x 1) = x is a linear equation. It can be converted to standard form. 1.2 Solving Linear Equations Equivalent Equations Two equations are equivalent if they have the same solutions. Example. x + 1 = 2, 2x = 2 and x = 1 are all equivalent equations. We can get an equivalent equation if we: Add or subtract each side of an equation by the same quantity. Multiply or divide each side of an equation by the same nonzero quantity. We may not multiply or divide by zero. Solve an equation by reducing it to a simpler equivalent form with an obvious solution. 2

3 Solving Linear Equations Problem. Solve the problem. Express your answer as an integer or simplified fraction. A B C D Solving Linear Equations 5x 7 5 = 7x Problem. Solve the formula for the specified variable. S = 2πrh + 2πr 2 for h A. h = S r B. h = S 2πr 1 C. h = S 2πr2 2πr D. h = 2π(S r) 3

4 2 Linear Inequalities 2.1 Linear Inequalities Linear Inequalities A linear inequality is one that can be written one of the following forms: ax + b < 0 (a 0) ax + b 0 (a 0) ax + b > 0 (a 0) ax + b 0 (a 0) 2.2 Interval Notation Interval Notation A double inequality a < x < b means that both a < x and x < b. This is the same as saying x is between a and b. Use interval notation to describe this set. An interval is: Closed if it contains its endpoints. Open if it doesn t contain any endpoints. Use [ and ] to denote included endpoints. Use ( and ) to denote excluded endpoints. 4

5 Interval Notation Example. The inequality 7 x < 5 may be written [ 7, 5) Interval [a, b] [a, b) (a, b] (a, b) (, b] (, b) [a, ) (a, ) (, ) Inequality a x b a x < b a < x b a < x < b x b x < b a x a < x R (, ) = R denotes the set of all real numbers. 2.3 Equivalent Inequalities Equivalent Inequalities Two inequalities are equivalent if they have the same solution sets. Example. The inequalities x > 1, x + 3 > 4, 2x > 2 and 3x < 3 are all equivalent. We can get an equivalent inequality if we: Add or subtract each side of an equation by the same quantity. Multiply or divide each side of an equation by the same nonzero quantity. Multiplying or dividing by a negative number changes the direction of the inequality. Multiplying or dividing by a positive number keeps the direction of the inequality unchanged. We may not multiply or divide by zero. Solve an inequality by reducing it to a simpler equivalent form with an obvious solution. 5

6 2.4 Solving Linear Inequalities Solving Linear Inequalities Problem. Solve and graph 6(2x 1) < 18x + 24 A. (, 3] < ] x B. (, 3) < ) x C. [3, ) [ > x D. (3, ) ( > x 3 Applications 3.1 Solving Word Problems Procedure for Solving Word Problems 1. Read the problem carefully and introduce a variable to represent an unknown quantity in the problem. 2. Identify other quantities in the problem (known or unknown) an express unknown quantities in terms of the variable you introduced in the first step. 3. Write a verbal statement using the conditions stated in the problem and then write an equivalent mathematical statement (equation or inequality). 4. Solve the equation or inequality and answer the questions posed in the problem. 5. Check that the solution solves the original problem. 6

7 3.2 Break-Even Analysis Break-Even Analysis Any manufacturing company has costs (C) and revenues (R). The company has a loss if R < C, a profit if R > C and will break-even if R = C. Costs may include fixed costs and variable costs Break-Even Analysis Problem. A publisher of a new math textbook figures that fixed costs will be $219, 975 and variable costs at $24.20 for each book produced. If the book is sold to distributors for $ each, how many must be sold for the publisher to break even? A. 81 books B books C books D books 4 Summary Summary You should be able to: Recognize and solve linear equations. Recognize and solve linear inequalities. Solve applications involving linear equations and inequalities. 7

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