Equations, Inequalities, and Problem Solving

Transcription

1 CHAPTER Equations, Inequalities, and Problem Solving. Linear Equations in One Variable. An Introduction to Problem Solving. Formulas and Problem Solving.4 Linear Inequalities and Problem Solving Integrated Review Linear Equations and Inequalities Today, it seems that most people in the world want to stay connected most of the time. In fact, 6% of U.S. citizens own cell phones. Also, computers with Internet access are just as important in our lives. Thus, the merging of these two into Wi-Fi-enabled cell phones might be the next big technological explosion. In Section., Objective, and Section., Exercises 5 and 6, you will find the projected increase in the number of Wi-Fi-enabled cell phones in the United States as well as the percent increase. (Source: Techcrunchies.com).5 Compound Inequalities.6 Absolute Value Equations.7 Absolute Value Inequalities Number of Wi-Fi-Enabled Cell Phones in the U.S. (in millions) Projected Growth of Wi-Fi-Enabled Cell Phones in the U.S. Mathematics is a tool for solving problems in such diverse fields as transportation, engineering, economics, medicine, business, and biology. We solve problems using mathematics by modeling real-world phenomena with mathematical equations or inequalities. Our ability to solve problems using mathematics, then, depends in part on our ability to solve equations and inequalities. In this chapter, we solve linear equations and inequalities in one variable and graph their solutions on number lines Year 47

2 4 CHAPTER Equations, Inequalities, and Problem Solving. Linear Equations in One Variable S Solve Linear Equations Using Properties of Equality. Solve Linear Equations That Can Be Simplified by Combining Like Terms. Solve Linear Equations Containing Fractions or Decimals. 4 Recognize Identities and Equations with No Solution. Solving Linear Equations Using Properties of Equality Linear equations model many real-life problems. For example, we can use a linear equation to calculate the increase in the number (in millions) of Wi-Fi-enabled cell phones. Wi-Fi-enabled cell phones let you carry your Internet access with you. There are already several of these smart phones available, and this technology will continue to expand. Predicted numbers of Wi-Fi-enabled cell phones in the United States for various years are shown below. Number of Wi-Fi-Enabled Cell Phones in the U.S. (in millions) Projected Growth of Wi-Fi-Enabled Cell Phones in the U.S Year 49 To find the projected increase in the number of Wi-Fi-enabled cell phones in the United States from 04 to 05, for example, we can use the equation below. Increase in cell phones in cell phones in In words: cell phones is 05 minus 04 Translate: x = 49 - Since our variable x (increase in Wi-Fi-enabled cell phones) is by itself on one side of the equation, we can find the value of x by simplifying the right side. x = The projected increase in the number of Wi-Fi-enabled cell phones from 04 to 05 is million. The equation x = 49 -, like every other equation, is a statement that two expressions are equal. Oftentimes, the unknown variable is not by itself on one side of the equation. In these cases, we will use properties of equality to write equivalent equations so that a solution may be found. This is called solving the equation. In this section, we concentrate on solving equations such as this one, called linear equations in one variable. Linear equations are also called first-degree equations since the exponent on the variable is. Linear Equations in One Variable x = y = y 4n - 9n + 6 = 0 z = -

3 Section. Linear Equations in One Variable 49 Linear Equations in One Variable A linear equation in one variable is an equation that can be written in the form ax + b = c where a, b, and c are real numbers and a 0. When a variable in an equation is replaced by a number and the resulting equation is true, then that number is called a solution of the equation. For example, is a solution of the equation x + 4 = 7, since + 4 = 7 is a true statement. But is not a solution of this equation, since + 4 = 7 is not a true statement. The solution set of an equation is the set of solutions of the equation. For example, the solution set of x + 4 = 7 is 56. To solve an equation is to find the solution set of an equation. Equations with the same solution set are called equivalent equations. For example, x + 4 = 7 x = x = are equivalent equations because they all have the same solution set, namely 56. To solve an equation in x, we start with the given equation and write a series of simpler equivalent equations until we obtain an equation of the form x number Two important properties are used to write equivalent equations. The Addition and Multiplication Properties of Equality If a, b, and c, are real numbers, then a = b and a + c = b + c are equivalent equations. Also, a = b and ac = bc are equivalent equations as long as c 0. The addition property of equality guarantees that the same number may be added to both sides of an equation, and the result is an equivalent equation. The multiplication property of equality guarantees that both sides of an equation may be multiplied by the same nonzero number, and the result is an equivalent equation. Because we define subtraction in terms of addition a - b = a + -b, and division in terms of multiplication a a b = a # b, these properties also guarantee that we may subtract the same b number from both sides of an equation, or divide both sides of an equation by the same nonzero number and the result is an equivalent equation. For example, to solve x + 5 = 9, use the addition and multiplication properties of equality to isolate x that is, to write an equivalent equation of the form x number We will do this in the next example. EXAMPLE Solve for x: x + 5 = 9. Solution First, use the addition property of equality and subtract 5 from both sides. We do this so that our only variable term, x, is by itself on one side of the equation. x + 5 = 9 x = 9-5 Subtract 5 from both sides. x = 4 Simplify. Now that the variable term is isolated, we can finish solving for x by using the multiplication property of equality and dividing both sides by. x = 4 Divide both sides by. x = Simplify.

4 50 CHAPTER Equations, Inequalities, and Problem Solving Check: To see that is the solution, replace x in the original equation with. x + 5 = 9 Original equation Let x = = 9 True Since we arrive at a true statement, is the solution or the solution set is 56. Solve for x: x + 7 =. Helpful Hint Don t forget that 0.4 = c and c = 0.4 are equivalent equations. We may solve an equation so that the variable is alone on either side of the equation. EXAMPLE Solve: 0.6 = -.5c. Solution We use both the addition property and the multiplication property of equality. 0.6 = -.5c = -.5c - Subtract from both sides. -.4 = -.5c Simplify. The variable term is now isolated = -.5c -.5 Divide both sides by = c Simplify Check: 0.6 = -.5c Replace c with Multiply. 0.6 = 0.6 True The solution is 0.4, or the solution set is Solve:.5 = -.5t. Solving Linear Equations That Can Be Simplified by Combining Like Terms Often, an equation can be simplified by removing any grouping symbols and combining any like terms. EXAMPLE Solve: -4x - + 5x = 9x + - 7x. Solution First we simplify both sides of this equation by combining like terms. Then, let s get variable terms on the same side of the equation by using the addition property of equality to subtract x from both sides. Next, we use this same property to add to both sides of the equation. -4x - + 5x = 9x + - 7x x - = x + Combine like terms. x - - x = x + - x Subtract x from both sides. -x - = Simplify. -x - + = + Add to both sides. -x = 4 Simplify. Notice that this equation is not solved for x since we have -x or -x, not x. To solve for x, we divide both sides by -. -x - = 4 - x = -4 Divide both sides by -. Simplify.

5 Section. Linear Equations in One Variable 5 Check to see that the solution is -4, or the solution set is Solve: -x x = 5x + - 4x. If an equation contains parentheses, use the distributive property to remove them. EXAMPLE 4 Solve: x - = 5x - 9. Solution First, use the distributive property. (x-)=5x-9 x - 6 = 5x - 9 Use the distributive property. Next, get variable terms on the same side of the equation by subtracting 5x from both sides. x - 6-5x = 5x - 9-5x Subtract 5x from both sides. -x - 6 = -9 Simplify. -x = Add 6 to both sides. -x = - Simplify. -x - = - Divide both sides by -. - x = Let x = in the original equation to see that is the solution. 4 Solve: x - 5 = 6x -. Solving Linear Equations Containing Fractions or Decimals If an equation contains fractions, we first clear the equation of fractions by multiplying both sides of the equation by the least common denominator (LCD) of all fractions in the equation. EXAMPLE 5 Solve for y: y - y 4 = 6. Solution First, clear the equation of fractions by multiplying both sides of the equation by, the LCD of denominators, 4, and 6. y - y 4 = 6 a y - y 4 b = a b Multiply both sides by the LCD. 6 a y b - a y 4 b = 4y - y = Apply the distributive property. Simplify. y = Simplify. Check: To check, let y = in the original equation. y - y 4 = 6 Original equation Let y =.

6 5 CHAPTER Equations, Inequalities, and Problem Solving Write fractions with the LCD. 6 Subtract. 6 = 6 Simplify. This is a true statement, so the solution is. 5 Solve for y: y - y 5 = 4. As a general guideline, the following steps may be used to solve a linear equation in one variable. Solving a Linear Equation in One Variable Step. Step. Step. Step 4. Step 5. Step 6. Clear the equation of fractions by multiplying both sides of the equation by the least common denominator (LCD) of all denominators in the equation. Use the distributive property to remove grouping symbols such as parentheses. Combine like terms on each side of the equation. Use the addition property of equality to rewrite the equation as an equivalent equation with variable terms on one side and numbers on the other side. Use the multiplication property of equality to isolate the variable. Check the proposed solution in the original equation. Helpful Hint When we multiply both sides of an equation by a number, the distributive property tells us that each term of the equation is multiplied by the number. EXAMPLE 6 Solve for x : x = x - x -. Solution Multiply both sides of the equation by, the LCD of and. a x + 5 a x b = ax - x - b Multiply both sides by. b + # = # x - x - a b Apply the distributive property. 4x = 6x - x - Simplify. 4x = 6x - x + Use the distributive property to remove parentheses. 4x + 4 = 5x + Combine like terms. -x + 4 = Subtract 5x from both sides. -x = - Subtract 4 from both sides. -x - = - - Divide both sides by -. x = Simplify. Check: To check, verify that replacing x with makes the original equation true. The solution is. 6 Solve for x: x - x - = x

7 Section. Linear Equations in One Variable 5 If an equation contains decimals, you may want to first clear the equation of decimals. EXAMPLE 7 Solve: 0.x + 0. = 0.7x Solution To clear this equation of decimals, we multiply both sides of the equation by 00. Recall that multiplying a number by 00 moves its decimal point two places to the right. 000.x + 0. = 000.7x x = 000.7x Use the distributive property. 0x + 0 = 7x - Multiply. 0x - 7x = Subtract 7x and 0 from both sides. x = - Simplify. x = - Divide both sides by. x = -4 Simplify. Check to see that the solution is Solve: 0.5x = 0.x CONCEPT CHECK Explain what is wrong with the following: x - 5 = 6 x = x = x = 4 Recognizing Identities and Equations with No Solution So far, each linear equation that we have solved has had a single solution. A linear equation in one variable that has exactly one solution is called a conditional equation. We will now look at two other types of equations: contradictions and identities. An equation in one variable that has no solution is called a contradiction, and an equation in one variable that has every number (for which the equation is defined) as a solution is called an identity. For review: A linear equation in one variable with No solution Is a Contradiction Every real number as a solution Is an Identity (as long as the equation is defined) The next examples show how to recognize contradictions and identities. Answer to Concept Check: Add 5 on the right side instead of subtracting 5. x - 5 = 6 x = x = 7 Therefore, the correct solution is 7. EXAMPLE Solve for x: x + 5 = x +. Solution First, use the distributive property and remove parentheses. x+5=(x+) x + 5 = x + 6 Apply the distributive property. x x = x x Subtract x from both sides. 5 = 6

8 54 CHAPTER Equations, Inequalities, and Problem Solving Helpful Hint A solution set of 506 and a solution set of 5 6 are not the same. The solution set 506 means solution, 0. The solution set 5 6 means no solution. The equation 5 = 6 is a false statement no matter what value the variable x might have. Thus, the original equation has no solution. Its solution set is written either as 5 6 or. This equation is a contradiction. Solve for x: 4x - = 4x + 5. EXAMPLE 9 Solve for x : 6x - 4 = + 6x -. Solution First, use the distributive property and remove parentheses. 6x-4=+6(x-) 6x - 4 = + 6x - 6 Apply the distributive property. 6x - 4 = 6x - 4 Combine like terms. At this point, we might notice that both sides of the equation are the same, so replacing x by any real number gives a true statement. Thus the solution set of this equation is the set of real numbers, and the equation is an identity. Continuing to solve 6x - 4 = 6x - 4, we eventually arrive at the same conclusion. 6x = 6x Add 4 to both sides. 6x = 6x Simplify. 6x - 6x = 6x - 6x Subtract 6x from both sides. 0 = 0 Simplify. Since 0 = 0 is a true statement for every value of x, all real numbers are solutions. The solution set is the set of all real numbers or, 5x x is a real number6, and the equation is called an identity. 9 Solve for x: 5x - = + 5x -. Helpful Hint For linear equations, any false statement such as 5 = 6, 0 =, or - = informs us that the original equation has no solution. Also, any true statement such as 0 = 0, =, or -5 = -5 informs us that the original equation is an identity. Vocabulary, Readiness & Video Check Use the choices below to fill in the blanks. Not all choices will be used. multiplication value like addition solution equivalent. Equations with the same solution set are called equations.. A value for the variable in an equation that makes the equation a true statement is called a(n) of the equation.. By the property of equality, y = - and y - 7 = are equivalent equations. 4. By the property of equality, y = - and y = - are equivalent equations. Identify each as an equation or an expression. 5. x x - = x + = 9 - x. 5 9 x x

9 Section. Linear Equations in One Variable 55 Martin-Gay Interactive Videos See Video. Watch the section lecture video and answer the following questions Complete these statements based on the lecture given before Example. The addition property of equality allows us to add the same number to (or subtract the same number from) of an equation and have an equivalent equation. The multiplication property of equality allows us to multiply (or divide) both sides of an equation by the nonzero number and have an equivalent equation. 0. From Example, if an equation is simplified by removing parentheses before the properties of equality are applied, what property is used?. In Example, what is the main reason given for first removing fractions from the equation?. Complete this statement based on Example 4. When solving a linear equation and all variable terms subtract out and: a. you have a statement, then the equation has all real numbers for which the equation is defined as solutions. b. you have a statement, then the equation has no solution.. Exercise Set Solve each equation and check. See Examples and.. -5x = -0. -x =. -0 = x = y x -. =.9 6. y -.6 = x - 4 = 6 + x. 5y - = + y z = x = -.. 5y + = y -. 4x + 4 = 6x + Solve each equation and check. See Examples and 4.. x - 4-5x = x x 4. x - 5x + = 4x x - 5x + = x x + x = -x x + = x x + = 7x x - 6 = 5x 0. 6x = 4x y - - y = -4y -. -4n - - n = -n - Solve each equation and check. See Examples 5 through x + x = 4 4. x + x 5 = 5 4 t 4 - t = 6. 4r 5 - r 0 = 7 n n + 5 = 5 + h. + h - = x - 0 =.4x x +.4 = 0.x x =. - x 5. 4n + = 6 + n 7. x = x + 9. x - + x = x x + 5 = x x MIXED Solve each equation. See Examples through x b = 5 + x = x Solve each equation. See Examples and n + 4 = + n x x = 6x x - + x = 7x x = 4x x - = x - + x a = x - 0 = -6x x - 7 = x x + x + 4 = 5x y + 0. = 0.6y w + 0. = w 5. 4 a + = 5 - a c = c y + 5y - 4 = 4y - y - 0 z = z + z x + = -0.x +

10 56 CHAPTER Equations, Inequalities, and Problem Solving 54. 9c - 6-5c = c - c x - x - = 4x x - x + 4 = x x x = -5x - - x x - - 0x = -x - 5-4x m m x - - x = 9x - 7 y = y y - - = y n4 + n = n + - n n n = n - 6n4 REVIEW AND PREVIEW Translating. Translate each phrase into an expression. Use the variable x to represent each unknown number. See Section The quotient of and a number 6. The sum of and a number 69. The product of and a number 70. The difference of and a number 7. Five subtracted from twice a number 7. Two more than three times a number CONCEPT EXTENSIONS Find the error for each proposed solution. Then correct the proposed solution. See the Concept Check in this section. 7. x + 9 = x = x = x = x +.6 = 4x x =. 5x 5 =. 5 x = 0.4 = y - - 4y = y x - 4 = n n -x - = 0 -x = x + 7 = 5x x + 7 = 5x 7 = 4x -x - = = 4x 4 7 = x = 5 6 x = - By inspection, decide which equations have no solution and which equations have all real numbers as solutions. 77. x + = x x - = 5x x + = x x - = 5x - 7. a. Simplify the expression 4x + +. b. Solve the equation 4x + + = -7. c. Explain the difference between solving an equation for a variable and simplifying an expression.. Explain why the multiplication property of equality does not include multiplying both sides of an equation by 0. (Hint: Write down a false statement and then multiply both sides by 0. Is the result true or false? What does this mean?). In your own words, explain why the equation x + 7 = x + 6 has no solution, while the solution set of the equation x + 7 = x + 7 contains all real numbers. 4. In your own words, explain why the equation x = -x has one solution namely, 0 while the solution set of the equation x = x is all real numbers. Find the value of K such that the equations are equivalent. 5..x + 4 = 5.4x - 7.x = 5.4x + K y - 0 = -.y y = -.y + K 7 x + 9 = x x = x + K x = x x + K = x 9. Write a linear equation in x whose only solution is Write an equation in x that has no solution. Solve the following. 9. xx = xx x + x - = 6xx x 9. xx = x + 0x xx = xx + 5 Solve and check x = y = z -.5 = x = -.5