Chapter 2: Linear Equations and Inequalities Lecture notes Math 1010

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1 Section 2.1: Linear Equations Definition of equation An equation is a statement that equates two algebraic expressions. Solving an equation involving a variable means finding all values of the variable for which the equation is true. Such values are solutions and are said to satisfy the equation. The solution set of an equation is the set of all solutions of the equation. If an equation has the set of all real numbers as its solution set, then it is called an identity. An equation whose solution set is not the entire set of real numbers is called a conditional equation. Ex.1 Equations. (1) 4x + 2 = 9 (2) 2x 4 = 2(x 2) (3) x 2 9 = 0 Ex.2 Checking a solution of an equation. Determine whether x = 3 is a solution of 3x 5 = 4x Properties of equalities Two equations that have the same set of solutions are equivalent equations. For example, x = 5 and x+5 = 0 are equivalent. An equation can be transformed into an equivalent equation using one or more of the following properties. (1) Simplify either side: remove symbols of grouping, combine like terms, or simplify fractions on one or both sides of the equation. (2) Apply the addition property equality: add or subtract the same quantity to/from each side of the equation. (3) Apply the multiplication property of equality: multiply or divide each side of the equation by the same nonzero quantity. (4) Interchange sides: interchange the two sides of the equation. 1

2 Ex.3 Solve 7 = 5x 2x + 1. Definition of linear equation A linear equation in one variable x is an equation that can be written in the standard form ax + b = 0, where a and b are real numbers with a 0. A linear equation in one variable is also called a first-degree equation because its variable has exponent 1. You can find the solution in the following way: Ex.4 Solve 4x 12 = 0. Then check the solution. 2

3 Ex.5 Solving an equation in nonstandard form. Solve 6(y 1) = 4y 2. Then check the solution. Ex.6 Solve x = 9. Ex.7 Solve 0.45x + 1.2(x 3) = 39. 3

4 Ex.8 Solve 2x 4 = 2(x 3). Ex.9 Solve 3x (x 6) = 5(x 2). 4

5 Section 2.2: Linear Equations and Problem Solving Ex.1 You have accepted a job at an annual salary of $40, 830. This salary includes a year-end bonus of $750. You are paid twice a month. What will your gross pay be for each paycheck? Write an algebraic expression that represents this problem. Then solve the equation and answer the question. Definition of percent A rate is a fraction that compares two quantities measured in different units. Rates that describe increases, decreases, and discounts are often given as percents. The word percent (= per cent) means divided by one hundred. Ex.2 (1) 10% = = 0.1 (2) 50% = = 1 2 = 0.5 (3) 25% = = 1 4 = 0.25 (4) 12.5% = = 1 8 =

6 Ex.3 The number 13 is 20% of what number? Ex.4 The number 28 is what percent of 80? 6

7 Section 2.4: Linear Inequalities Algebraic inequalities Algebraic inequalities are inequalities that contain one or more variable terms. For example, x < 3, x 5 and x 2 > 2x + 5 are algebraic inequalities. Solving an inequality in the variable x means that you need to find the values of x for which the inequality is true. Those values are the solutions of the inequality and the set of all solutions is called solution set. The graph of an inequality is the plot of the solutions in the real line. Bounded and unbounded intervals Let a and b be real numbers such that a < b. The following intervals are called bounded intervals: The length of intervals [a, b], [a, b), (a, b] and (a, b) is b a. If an interval doesn t have a finite length, then it is called unbounded. The following intervals are unbounded: 7

8 Ex.1 Sketch the graph of each inequality. (1) 5 < x < 8 (2) x 1 (3) 4 x 6 (4) x 2 Properties of inequalities Let a, b, and c be real numbers, variables, or expressions. (1) Addition and subtraction properties Adding the same quantity to (subtracting the same quantity from) each side of an inequality produces an equivalent inequality (that is an inequality with the same solution set). If a < b, then a + c < b + c. If a < b, then a c < b c. (2) Multiplication and division properties Multiplying (dividing) each side of an inequality by a positive quantity produces an equivalent inequality. Multiplying (dividing) each side of an inequality by a negative quantity produces an equivalent inequality in which the inequality symbol is reversed. If a < b, and c > 0 then ac < bc. If a < b, and c > 0, then a c < b c. If a < b, and c < 0 then ac > bc. If a < b, and c < 0, then a c > b c. (3) Transitive property If a < b and b < c, then a < c. Linear inequalities An inequality is called linear if it has one of the following forms: ax + b 0, ax + b < 0, ax + b 0, ax + b > 0 8

9 Ex.2 Solve x + 4 < 9. Ex.3 Solve 12 4x 30. Ex.4 Solve 7x 3 > 3(x + 1). Ex.5 Solve 2x < x

10 Compound inequalities Two inequalities joined by the word and or the word or give a compound inequality. When two inequalities are joined by the word and, the solution set consists of all real numbers that satisfy both inequalities. When two inequalities are joined by the word or, the solution set consists of all real numbers that satisfy one of the two inequalities. A compound inequality formed by the word and is called conjunctive and a compound inequality formed by the word or is called disjunctive. Ex.6 Solve 7 5x 2 < 8. 10

11 Ex.7 Solve the compound inequality 1 2x 3 and 2x 3 < 5. Ex.8 Solve the compound inequality 1 2x 3 or 2x 3 < 5. Ex.9 A solution set is (1) Write the solution set as a compound inequality. (2) Write the solution set using the union symbol. 11

12 Ex.10 Write the compound inequality 3 x 5 using the intersection symbol. Ex.11 (1) x is at most n: x n. (2) x is no more that n: x n. (3) x is at least n: x n. (4) x is no less than n: x n. (5) x is more than n: x > n. (6) x is less than n: x < n. (7) x is a minimum of n: x n. (8) x is at least m, but less than n: m x < n. (9) x is greater than m, but no more than n: m < x n. 12

13 Section 2.5: Absolute Value Equations and Inequalities Solving absolute value equation Let x be a variable or an algebraic expression and let a be a real number such that a 0. The solutions of the equation x = a are x = a and x = a. Ex.1 Solve each absolute value equation. (1) x = 12 (2) y = 3 (3) z = 0 (4) x = 3 13

14 Ex.2 Solve 3x + 4 = 10. Ex.3 Solve 2x = 8. Ex.4 Solve 3x 4 = 7x

15 Ex.5 Solve x + 5 = x Solving absolute value inequality Let x be a variable or an algebraic expression and let a be a real number such that a > 0. (1) The solutions of x < a are values of x such that a < x < a. (2) The solutions of x > a are values of x such that x < a or x > a. Ex.6 Solve x 5 < 2. 15

16 Ex.7 Solve 3x 4 5. Ex.8 Solve 2 x 3 <