Free Pre-Algebra Lesson 24 page 1

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1 Free Pre-Algebra Lesson page 1 Lesson Equations with Negatives You ve worked with equations for a while now, and including negative numbers doesn t really change any of the rules. Everything you ve already learned is still valid. But it s still nice to go through some examples carefully and catch the little details that sometimes feel tricky in this new situation. Review About Equations Now that you ve solved equations for a few weeks, let s review exactly what we re doing. With your added experience, you ll have insights you perhaps didn t the first time. An equation is a kind of mathematical sentence, and the equals sign is like the verb. In fact, when translating words to algebra, we usually translate the word is to an equals sign. An equals sign is a very, very strong symbol in mathematics. Many students think of it as a sort of and next symbol, since it often comes between the steps of a problem. But it is much stronger than that it means that the things it is linking are identical and interchangeable, merely disguised versions of one another. Let s look at an example to help a little with clarity in this abstract discussion. Here is an equation: x 1 7 The equals sign separates the two sides of the equation, and indicates that they are EQUAL, that is, they are exactly the same, mathematically speaking, because they represent the SAME AMOUNT. Here the two sides are x 1 and 7. To solve the equation means to find any numbers that can replace x to make the equation true, that is, to make the two sides equal. Those numbers are called solutions to the equation. The procedure for finding solutions is to strip away the operations that were done to the variable by doing the opposite operations in the reverse order on both sides of the equals sign. Because the two sides begin equal and have exactly the same experiences, they stay equal to one another. The goal is to have the variable alone on one side of the equal sign (sometimes called isolating the variable) and any solutions on the other. x 1 7 x x 8 x / 8 / x The solution can replace x in the original equation and the two sides are equal: x x 1 7 ()

2 Free Pre-Algebra Lesson page Equations with Only One Operation We start with simple equations to see how to deal with negatives in each of the four arithmetic operations. Adding or Subtracting If a negative number is added to or subtracted from the variable, simplify first, then proceed as formerly. Example: Solve the equation x + ( 3) = 5. First simplify by re-writing the equation as a subtraction: x 3 = 5 Then solve as usual by doing the opposite operation on both sides. x 3 5 x x 8 Example: Solve the equation x ( 7) =. First simplify by re-writing the equation as an addition: x + 7 = Then solve as usual by doing the opposite operation on both sides. The solution to this equation is a negative number. x 7 x x 5 Multiplying or Dividing When undoing a multiplication or division with a negative number, you must do the opposite operation with the same negative number. Example: Solve the equation 3x = 5. In order to undo the operation of multiplying by 3 we must divide by 3, because 3 / ( 3) = 1. 3x 5 3x / ( 3) 5 / ( 3) x 15 Example: Solve the equation x /(-) =. To cancel the in the denominator, both sides must be multiplied by. Another negative number solution. x x ( ) ( ) x 8 Multi-Operation Equations with Negatives This is just practice. Nothing is different, or new, now that you know how to deal with a negative number in each operation. Example: Solve the equations. 5x x x x 30 5x / ( 5) 30 / ( 5) x 6 x 7 6 x 7 x x 7 1 x x 5

3 Free Pre-Algebra Lesson page 3 Dealing with x One way to deal with x is to remember that x is just shorthand for 1x. Example: Solve the equations. x 5 1x 5 1x / ( 1) 5 / ( 1) x 5 x 5 1x 5 1x / ( 1) 5 / ( 1) x 5 Once you ve gone through these steps a few times it begins to feel like overkill. A slightly shorter thought process occurs if you interpret x as the opposite of x. Then you can take the opposite of both sides of the equation without writing a division. You should use whichever of these method makes the most sense to you, or any other method that you understand and that works. (Some people multiply both sides of the equation by 1, which has the same result.) Just remember that if your solution says x instead of x, you re not quite done with the problem yet, and have one more simplification step. Example: Solve the equation 9 x =. 9 x It s important to remember that the subtraction sign is applied to the x, not the 9. If we re-write the equation to change the subtraction to addition, you can see it more clearly. Each of the gray equations to the right is equivalent to the equation above. The last version certainly looks like something we already know how to solve. 1x 9 1x x 7 1x / ( 1) / ( 1) x 7 9 x 9 1x 1x 9 If we want to solve the original equation without rearranging and using the take the opposite shortcut, it looks like this: 9 x 9 x 9 9 x 7 x 7 The tricky part is to subtract 9 from both sides, because the 9 is a positive number added to x. Many people see the sign and think they need to add 9, but the minus applies to the x. Check: x 7 9 x 9 (7)

4 Free Pre-Algebra Lesson page Example: Solve the equation 6 x =. Rewrite the problem by changing the subtraction to + x or to + 1x. Option 1 Option 6 x 6 x x 6 x 6 6 x 6 1x 1x 6 1x 6 Then solve the equation in its new form. Option 1 x 6 x x 8 x 8 Check: x 8 6 x 6 ( 8) 6 8 true Option 1x 6 1x x 8 1x / ( 1) 8 / ( 1) x 8

5 Free Pre-Algebra Lesson page 5 Negative Cheat Sheet Type 1 Models Type Models Comparing with > and < Any negative number is less than any positive number It is worse to owe $100 than to owe $99, or, 100º is colder than 99º, etc. Addition Battle or Party? Battles Absolute Value: makes everything positive: Also, 0 0 gives priority like parentheses in the order of operations: Parties Subtraction Change to addition. Multiplication & Division Like signs, positive, unlike signs, negative. Exponents Write out the multiplication. Fractions are divisions. Algebraic Expressions have super-efficient notation. Formulas Parentheses around numbers when substituting! Equations It can help to convert x to 1x before solving ( 5) / / / / 5 3 Be careful with subtle notation differences: compare the multiplication 3( 5) to the subtraction 3 ( 5) ( 3) ( 3)( 3) 9 3 (3 ) (3 3) 9 ( 3) 3 ( 3)( 3)( 3) (3 3 ) (3 3 3) 7 8 ( 8) 8 ( 8) ( ) ( ) x 5x x 3x x 1x x 3x 3x 0x 0 3x x 1x x 3(x 1) 6x ( 3) 6x 3 3(x 1) 3(x ( 1)) 6x 3 t 3 h 16t 80t 6 x 3 x x h 16(3) 80(3) ( 3) ( 3) Interpret negative answers in the context of the problem (loss vs. profit, below vs. above ground, etc.) To solve 3 x change to 1x 3 1x If x 5, then x 5. 1x 7 1x / ( 1) 7 / ( 1) If x 5, then x 5. x 7

6 Free Pre-Algebra Lesson page 6 Lesson : Equations with Negatives Worksheet Name Solve the equations. The answers may be positive or negative whole numbers or fractions. 8a 5 13 b (c 9) 1 d 5 k 7 1 3m 5 7 3x x 18 x 16 Choose one of the problems above and check the answer in the original equation.

7 Free Pre-Algebra Lesson page 7 Lesson : Equations with Negatives Homework A Name 1. Translate the words to mathmatical notation. a. It is better to owe $0 than to owe $100. b. The product of 1 and is. c. When 5 positive particles are combined with negative particles the result is that 1 positive particle remains.. Find the absolute value: a. 5 b. 0 c. 9 d. ( 1) 3. Compare with > or <. a. 7 0 b. 7 3 c d Compare with <, >, or =. a b. (7 1) (1 7) c. (7)( 1) ( 1)( 7) 5. Simplify ( 8) 9 ( 8) 9 ( 8) 9 / ( 1) 9 ( 1) 5 ( 5) ( 1)( )( 3) ( 5) Simplify the algebraic expressions. Use super-efficient notation in your answers. x 8 5x 9 6x ( 5x) 9x 8y 7x (y)

8 Free Pre-Algebra Lesson page 8 7. Use the given formula to solve the problem. The height above ground in feet of a small model rocket fired from ground level (disregarding air resistance) is given by the equation h 16t 19t Convert 00ºC to Fahrenheit using the formula 9C F 3 5 The rocket reaches its greatest height after 6 seconds. What is the height when t = 6? Convert 193ºF to Celsius using the formula What is the height when t = 1? C 5(F 3) 9 8. Evaluate each expression for the given values of the variable. x x x x 9. Solve the equations. x x 7

9 Free Pre-Algebra Lesson page 9 Lesson : Equations with Negatives Homework A Answers 1. Translate the words to mathmatical notation. a. It is better to owe $0 than to owe $ b. The product of 1 and is. ( 1)() c. When 5 positive particles are combined with negative particles the result is that 1 positive particle remains Find the absolute value: a. 5 5 b. 0 0 c. 9 9 d. ( 1) 1 3. Compare with > or <. a. 7 < 0 b. 7 < 3 c. 7 > 10 d. 7 < 10. Compare with <, >, or =. a b. (7 1) (1 7) 6 6 c. (7)( 1) ( 1)( 7) Simplify ( 8) 9 ( 8) 9 ( 8) / ( 1) 9 ( 1) 5 (5 5) 5 ( 5) ( 5)( 5) ( 1)( )( 3) ()( 3) ( 5) (9 1) 10 (10 10) 6. Simplify the algebraic expressions. Use super-efficient notation in your answers. x 8 5x x 5x 8 1x 8 x 8 9 6x ( 5x) 9 6x 5x 9 1x 9 x 9x 8y 7x (y) 9x 7x 8y 8y x 0y x

10 Free Pre-Algebra Lesson page Use the given formula to solve the problem. The height above ground in feet of a small model rocket fired from ground level (disregarding air resistance) is given by the equation h 16t 19t The rocket reaches its greatest height after 6 seconds. What is the height when t = 6? h 16(6) 19(6) 16(36) 19(6) feet above ground What is the height when t = 1? h 16(1) 19(1) 16(1) 19(1) The rocket hits ground. Convert 00ºC to Fahrenheit using the formula C 00 F 0 9( 00 ) 5 F 9C ºF Convert 193ºF to Celsius using the formula F 193 C C 5(( 193) 3) 9 5(F 3) 9 5 5( 5 ) 9 15ºC 8. Evaluate each expression for the given values of the variable. x x x x 9. Solve the equations. x 9 3 x 9 3 x x 6 x / 6 / 6 3 x 6 x 7 x 6 7 x x 6 x / 6 / 6 3 x

11 Free Pre-Algebra Lesson page 11 Lesson : Equations with Negatives Homework B Name 1. Translate the words to mathmatical notation. a. Owing $75 is better than owing $100.. Find the absolute value: a. 0 b Compare with > or <. a. 9 7 b Compare with <, >, or =. a ( 1) b ( 9) b. The quotient of and is 1. c. The checking account had $30, but there was a debit of $50, resulting in an overdraft of $0. c. 3 d. 0 5 c. 9 9 d c ( 9) 5. Simplify. ( ) / ( ) ( ) ( ) ( )( 3)( ) ( 3) 6. Simplify the algebraic expressions. Use super-efficient notation in your answers. 3x 7x x 6y 5y 3 3(x )

12 Free Pre-Algebra Lesson page 1 7. Use the given formula to solve the problem. The height above ground in feet of a small model rocket fired from ground level (disregarding air resistance) is given by the equation h 16t t Convert 300ºC to Fahrenheit using the formula 9C F 3 5 The rocket reaches its greatest height after 7 seconds. What is the height when t = 7? What is the height when t = 1? Convert 301ºF to Celsius using the formula 5(F 3) C 9 8. Evaluate each expression for the given values of the variable. 3 3x 3x 3 9. Solve the equations. 8y x