Chapter 2 Linear Equations and Inequalities in One Variable

Transcription

1 Chapter 2 Linear Equations and Inequalities in One Variable

2 Section 2.1: Linear Equations in One Variable Section 2.3: Solving Formulas Section 2.5: Linear Inequalities in One Variable Section 2.6: Compound Inequalities in One Variable Section 2.7: Absolute Value Equations and Inequalities

3 Section 2.1: Linear Equations in One Variable An equation is a mathematical statement that two expressions are equal. An equation contains an = sign and an expression does not. 5x + 4 = 7 equation 9y + 2 expression We can solve equations, and we can simplify expressions. To solve an equation means to find the value(s) of the variable that make the equation true.

4 Linear Equation in One Variable A linear equation in one variable is an equation that can be written in the form ax + b = 0 where a and b are real numbers and a 0. Notice that the exponent of the variable (x) is 1 in a linear equation. For this reason, these equations are also known as first-degree equations. Here are other examples of linear equations in one variable: 5y 8 = 19 2(k + 7) 3 = 4k t = t 5

5 The properties of equality will help us solve equations. The Properties of Equality Let a, b, and c be expressions representing real numbers. Then, If a = b, then a + c = b + c. If a = b, then a c = b c. If a = b, then ac = bc. If a = b, then a c = b (c 0). c Addition Property Subtraction Property Multiplication Property Division Property These properties tell us that we can add, subtract, multiply, or divide both sides of an equation by the same real number without changing the solutions to the equation.

6 Examples x 8 = 5 9y = t = 20

7 Sometimes it is necessary to combine like terms before we apply the properties of equality. Here are the steps we use to solve a linear equation in one variable. How to Solve a Linear Equation (1) Clear parentheses and combine like terms on each side of the equation. (2) Get the variable on one side of the equal sign and the constant on the other side of the equal sign (isolate the variable) using the addition or subtraction property of equality. (3) Solve for the variable using the multiplication or division property of equality. (4) Check the solution in the original equation.

8 Example Solve: 5(3 2x) + 7x 2 = 2(9 2x) + 15

9 Some equations contain several fractions or decimals which make them appear more difficult to solve. Here are two examples: 2 9 x 1 2 = 1 18 x and 0.05y + 0.4(y 3) = 0.3 Before applying the steps for solving a linear equation, we can eliminate the fractions and decimals from the equations. Eliminating Fractions from an Equation To eliminate the fractions, determine the least common denominator (LCD) for all of the fractions in the equation. Then, multiply both sides of the equation by the LCD.

10 Example Solve: 1 5 y + 1 = 3 10 y + 1 4

11 Just as we can eliminate the fractions from an equation to make it easier to solve, we can eliminate decimals from an equation to make it easier to solve. Eliminating Decimals from an Equation To eliminate the decimals from an equation, multiply both sides of the equation by the smallest power of 10 that will eliminate all decimals from the problem.

12 Example Solve: 0.1a = (a 5)

13 Outcomes When Solving Linear Equations There are three possible outcomes when solving a linear equation. The equation may have (1) one solution. Solution set: a real number. (2) no solution. In this case, the variable will drop out, and there will be a false statement such as 2 = 5. Solution set:. (3) an infinite number of solutions. In this case, the variable will drop out, and there will be a true statement such as 8 = 8. Solution set: R.

14 Examples Solve: 6 + 5x 4 = 3x x 3x 4x + 9 = 5 x

15 Section 2.3: Solving Formulas The formula P = 2l + 2w allows us to find the perimeter of a rectangle when we know its length, l, and width, w. But, what if we were solving problems where we repeatedly needed to find the value of w? Then, we could rewrite P = 2l + 2w so that it is solved for w: w = P 2l 2 Doing this means that we have solved the formula P = 2l + 2w for the specific variable, w. Solving a formula for a specific variable may seem confusing at first because the formula contains more than one letter. Keep in mind that we will solve for a specific variable the same way we have been solving equations up to this point.

16 Examples Solve 2x + 5 = 13 and mx + b = y for x. Solve R = ρl A for L. Solve A = 1 h (b1 + b2) for b1. 2

17 Section 2.5: Linear Inequalities in One Variable Recall the inequality symbols < is less than > is greater than is less than or equal to is greater than or equal to We will use the symbols to form linear inequalities in one variable. While an equation states that two expressions are equal, an inequality states that two expressions are not necessarily equal. Here is a comparison of an equation and an inequality: Equation Inequality 3x 8 = 13 3x 8 13

18 Linear Inequality in One Variable A linear inequality in one variable can be written in one of the following forms ax + b < c ax + b > c ax + b c ax + b c where a, b, and c are real numbers and a 0. The solution to a linear inequality is a set of numbers that can be represented in one of three ways: On a graph In set notation In interval notation In this section, we will learn how to solve linear inequalities in one variable and how to represent the solution in each of those three ways.

19 Examples Graph each inequality and express the solution in set notation and interval notation. k 7 x < 5

20 The addition and subtraction properties of equality help us to solve equations. Similar properties hold for inequalities as well. Addition and Subtraction Properties of Inequality Let a, b, and c be real numbers. Then, a < b and a + c < b + c are equivalent a < b and a c < b c are equivalent Adding the same number to both sides of an inequality or subtracting the same number from both sides of an inequality will not change the solution. The above properties hold for any of the inequality symbols.

21 Example Solve y Graph the solution set and write the answer in interval and set notations.

22 While the addition and subtraction properties for solving equations and inequalities work the same way, this is not true for multiplication and division. Multiplication and Division Properties of Inequality Let a, b, and c be real numbers. If c is a positive number, then a < b and ac < bc are equivalent inequalities and have the same solutions. a < b and a c < b are equivalent inequalities and have the same solutions. c If c is a negative number, then a < b and ac > bc are equivalent inequalities and have the same solutions. a < b and a c > b are equivalent inequalities and have the same solutions. c

23 Examples Solve each inequality. Graph the solution set and write the answer in interval and set notations. 5x y > 3

24 Example Often it is necessary to combine the properties to solve an inequality. Solve 4(k + 2) + 1 2(3k + 10). Graph the solution set and write the answer in interval and set notations.

25 A compound inequality contains more than one inequality symbol. Some types of compound inequalities are 5 < b + 4 < 1 t 1 2 or t 3 z < 2 and z > 7 Consider the inequality 2 x 3. We can think of this in two ways: x is between 2 and 3, and 2 and 3 are included in the interval. We can break up 2 x 3 into the two inequalities x 2 and x 3. Either way we think about 2 x 3, the meaning is the same. On a number line, the inequality would be represented as Notice that the lower bound of the interval on the number line is 2 (including 2), and the upper bound is 3 (including 3). Therefore, we can write the interval notation as [ 2, 3] The set notation to represent 2 x 3 is {x 2 x 3}.

26 Examples Solve each inequality. Graph the solution set, and write the answer in interval notation. 2 < 7k 9 < < 1 3 x y 4 < 10

27 Section 2.6: Compound Inequalities in One Variable In Section 2.5, we learned how to solve a compound inequality like 8 3x In this section, we will discuss how to solve compound inequalities like these: t 1 2 or t 3 2z + 9 < 5 and z 1 > 6 But first, we must talk about set notation and operations.

28 Let A = {1, 2, 3, 4, 5, 6} and B = {3, 5, 7, 9, 11}. The intersection of sets A and B is the set of numbers that are elements of A and of B. The intersection of A and B is denoted by A B A B = {3, 5} since 3 and 5 are found in both A and B. The union of sets A and B is the set of numbers that are elements of A or of B. The union of A and B is denoted by A B. The set A B consists of the elements in A or in B or in both. A B = {1, 2, 3, 4, 5, 6, 7, 9, 11} Although the elements 3 and 5 appear in both set A and in set B, we do not write them twice in the set. The word and indicates intersection, while the word or indicates union. This same principle holds when solving compound inequalities involving and or or.

29 Steps for Solving a Compound Inequality (1) Identify the inequality as and or or and understand what that means. (2) Solve each inequality separately. (3) Graph the solution set to each inequality on its own number line even if the problem does not explicitly tell you to graph the solution set. This will help you to visualize the solution to the compound inequality. (4) Use the separate number lines to graph the solution set of the compound inequality. (a) If it is an and inequality, the solution set consists of the regions on the separate number lines that would overlap (intersect) if one number line was placed on top of the other. (b) If it is an or inequality, the solution set consists of the total (union) of what would be shaded if you took the separate number lines and put one on top of the other. (5) Use the graph of the solution set to write the answer in interval notation.

30 Examples Solve each compound inequality and write the answer in interval notation. 4 x > 8 and 2x t or 3 2 t < 6

31 Examples There are some special compound inequalities that can occur. Solve each compound inequality and write the answer in interval notation. 3y y 6 and 5y < 4 9k 8 8 or k + 7 2

32 Section 2.7: Absolute Value Equations and Inequalities The absolute value of a number describes its distance from zero. 5 = 5 and 5 = 5 We use this idea of distance from zero to solve absolute value equations and inequalities.

33 Solving an Absolute Value Equation If P represents an expression and k is a positive real number, then to solve P = k we rewrite the absolute value equation as the compound equation P = k or P = k and solve for the variable. P can represent expressions such as x 3a t 9

34 Examples Solve each equation. x 4 = k = 5

35 Example Recall that an absolute value results in a non-negative number. In the event that an absolute value is equal to a negative number, the equation has no solution. The solution set is. Solve t + 3 = 5

36 Another type of absolute value equation involves two absolute values. Solve ax + b = cx + d If P and Q are expressions, then to solve P = Q we rewrite the absolute value equation as the compound equation. and solve for the variable. P = Q or P = Q

37 Example Solve x + 7 = 3x 1

38 Next, we will learn how to solve absolute value inequalities. Some examples of absolute value inequalities are t < 6 x k 1 > y 3 What does it mean to solve x 3? It means to find the set of all real numbers whose distance from zero is 3 units or less. Any number between 3 and 3 is less than 3 units from zero. We can represent the solution set on a number line as We can write the solution set in interval notation as [ 3, 3]. Solve P < k or P k Let P be an expression and let k be a positive, real number. To solve P < k, solve the compound inequality k < P < k. (similarly for )

39 Examples Solve each inequality. Graph the solution set and write the answer in interval notation. 4 5x < 16 6k

40 To solve x 4 means to find the set of all real numbers whose distance from zero is 4 units or more. Any number greater than 4 or less than 4 is more than 4 units from zero. We can represent the solution set to x 4 as The solution set consists of two separate regions, so we can write a compound inequality using or. x 4 or x 4 In interval notation, we write (, 4] [4, ). Solve P > k or P k Let P be an expression and let k be a positive, real number. To solve P > k, solve the compound inequality P < k or P > k. (similarly for )

41 Examples Solve each inequality. Graph the solution set and write the answer in interval notation. 8y k > 9

42 Examples There are some special absolute value inequalities that can occur. Solve each compound inequality and write the answer in interval notation. x y 7 < 2 6k