Equations & Inequalities Grade 6 Copyright © Ed2Net Learning, Inc.

Writing Expressions & Equations Addition Subtraction Multiplication Division Plus The sum of Increased by Total More than Added to Minus The difference of Decreased by Fewer than Less than Subtracted from Times The product of Multiplied by Of Divided by The quotient of Copyright © Ed2Net Learning, Inc.

Writing Expressions & Equations Phrase Expression The sum of 4 and a number 4 + x The quotient of a number and 6 y/6 Sentence Expression A number x minus 5 is 12 x – 5 = 12 15 times a number y is 75 15y = 75 Copyright © Ed2Net Learning, Inc.

Solving Addition Equations NOTE BOOK Solving Addition Equations To solve an addition equation, subtract the same number from each side so that the variable is by itself on one side. Copyright © Ed2Net Learning, Inc.

Copyright © Ed2Net Learning, Inc. Solving Addition Equations EXAMPLE 1 Solving an Addition Equation Solve the equation y + 25 = 140. y + 25 = 140 Write the original equation. ––––– – 25 –––– – 25 Subtract 25 from each side. y = 115 Simplify. Copyright © Ed2Net Learning, Inc.

Copyright © Ed2Net Learning, Inc. Solving Addition Equations EXAMPLE 2 Solving an Addition Equation Solve the equation y + 25 = 140. y + 25 = 140 Write the original equation. ––––– – 25 –––– – 25 Subtract 25 from each side. y = 115 Simplify. CHECK y + 25 = 140 Write the original equation. y + 25 = ? 140 115 Substitute 115 for y. 140 = 140 Solution checks. Copyright © Ed2Net Learning, Inc.

Copyright © Ed2Net Learning, Inc. Solving Addition Equations EXAMPLE 3 Using an Addition Equation Shopping You buy some clothing that costs $17.45. What is the amount of change c that the clerk should give you if you pay with a $20 bill? SOLUTION Cost + Change = Amount paid Write a verbal model. 17.45 + c = 20.00 Write an equation. ––––––––– –17.45 –––––– –17.45 Subtract 17.45 from each side. c = 2.55 Simplify. ANSWER The clerk should give you $2.55 in change. Copyright © Ed2Net Learning, Inc.

Solving Subtraction Equations To solve a subtraction equation, add the same number to each side so that the variable is by itself on one side. Solve x – 2 = 5 Add 2 to both sides: x – 2 + 2 = 5 + 2 x = 7 Copyright © Ed2Net Learning, Inc.

Using a Subtraction Equation Example: You are riding an elevator. You go down 14 floors and exit on the 23rd floor. On what floor did you enter the elevator? Let F represent the number of the floor on which you entered the elevator. F – 14 = 23 F – 14 + 14 = 23 + 14 F = 37 You entered the elevator on the 37th floor. Copyright © Ed2Net Learning, Inc.

Solving Multiplication & Division Equations To solve a multiplication equation, divide each side by the number the variable is multiplied by. To solve a division equation, multiply each side by the divisor. Copyright © Ed2Net Learning, Inc.

Solving a Multiplication Equation Solve 5x = 20 5x/5 = 20/5 x = 4 Divide each side by 5. Copyright © Ed2Net Learning, Inc.

Solving a Division Equation Solve x/7 = 3 x/7 = 3 7.x/7 = 7.3 x = 21 Divide each side by 5. Copyright © Ed2Net Learning, Inc.

Copyright © Ed2Net Learning, Inc. Using an Equation Example: In cheerleader tryouts, 27 students are placed in groups of 3 to make human pyramids. Write and solve a multiplication equation to find n, the number of pyramids the 27 students form. 27 = 3n 27/3 = 3n/3 9 = n The 27 students will form 9 pyramids. Copyright © Ed2Net Learning, Inc.

Copyright © Ed2Net Learning, Inc. Solving Inequalities An inequality is a statement formed by placing an inequality symbol between two expressions. To translate sentences into inequalities, look for the following phrases. Phrases Symbol Is less than < Is less than or equal to < Is greater than > Is greater than or equal to > Copyright © Ed2Net Learning, Inc.

Writing Simple Inequalities Sentence Inequality A number is less than 5 x < 5 Twice a number is greater than or equal to 8. 2x > 8 A number minus 7 is less than or equal to 5. x – 7 < 5 Copyright © Ed2Net Learning, Inc.

Copyright © Ed2Net Learning, Inc. Solving Inequalities The solution of an inequality is the set of all values of the variable that make the inequality true. Solving an inequality is similar to solving an equation. Copyright © Ed2Net Learning, Inc.

Copyright © Ed2Net Learning, Inc. Solving Inequalities Example: Solve x + 2 > 3 x + 2 – 2 > 3 – 2 x > 1 This means that the solution is the set of all numbers that are greater than or equal to 1. Copyright © Ed2Net Learning, Inc.

Graphing Solutions of an Inequality The graph of an inequality is all the points on a number line that represent the solution of the inequality. An open dot on a graph indicates a number that is not part of the solution. Copyright © Ed2Net Learning, Inc.

Graphing Solutions of an Inequality Solve and then graph 3x < 21 3x / 3 < 21/3 x < 7 7 is not part of the solution, so use an open dot at 7 on the graph. -1 0 1 2 3 4 5 6 7 8 9 10 Copyright © Ed2Net Learning, Inc.

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Copyright © Ed2Net Learning, Inc. Functions In the Real World Giant Pandas Giant pandas eat about 30 pounds of bamboo every day. About how many pounds of bamboo will a giant panda eat in 2 days? in 3 days? in 4 days? The function rule below relates the pounds of bamboo a panda eats to the number of days. Pounds of Bamboo = 30  Number of days A function is a pairing of each number in one set with a number in a second set. Starting with a number in the first set, called an input, the function pairs it with exactly one number in the second set, called an output. Copyright © Ed2Net Learning, Inc.

Substitute in the function p = 30d Functions EXAMPLE 1 Evaluating a Function To solve the problem about about giant pandas, you can make an input-output table. Use the function rule p = 30d, where d is the number of days (input) and p is the pounds of bamboo eaten (output). Input Days, d Substitute in the function p = 30d Output Pounds eaten, p 1 p = 30(1) 30 2 p = 30(2) 60 3 p = 30(3) 90 4 p = 30(4) 120 A giant panda will eat about 60 pounds of bamboo in 2 days, about 90 pounds in 3 days, and about 120 pounds in 4 days. ANSWER Copyright © Ed2Net Learning, Inc.

Copyright © Ed2Net Learning, Inc. Functions EXAMPLE 2 Using a Table to Write a Rule Write a function rule for the input-output table. Input, x Output, y 10 6 11 7 12 8 13 9 SOLUTION Each output y is 4 less than the input x. A function rule is y = x – 4. Copyright © Ed2Net Learning, Inc.

Copyright © Ed2Net Learning, Inc. Functions EXAMPLE 2 Using a Table to Write a Rule Write a function rule for the input-output table. Input, x Output, y Input, m Output, n 10 6 11 7 3 1 12 8 6 2 13 9 9 3 SOLUTION SOLUTION Each output y is 4 less than the input x. A function rule is y = x – 4. m 3 Each output n is the input m divided by 3. A function rule is n = . Copyright © Ed2Net Learning, Inc.

Copyright © Ed2Net Learning, Inc. Functions EXAMPLE 3 Making a Table to Write a Rule Pattern Make an input-output table using the number of squares s as the input and the number of triangles t as the output. Then write a function that relates s and t. 1 2 3 4 Each output value is 2 times the input value.There are 2 triangles for every square. Squares, s Triangles, t SOLUTION 1 2 2 4 3 6 ANSWER A function rule for this pattern is t = 2s. 4 8 Copyright © Ed2Net Learning, Inc.

Copyright © Ed2Net Learning, Inc. Graphing Functions In the Real World Walking You are training for a long distance walking race. In your practice walks, you maintain a steady rate of about 15 minutes per mile. How can you use a graph to represent this relationship? The number of miles you walk x and the number of minutes it takes y are related by the rule y = 15x. So, the distances and times for practice walks are represented by the points on the graph of the function y = 15x. Copyright © Ed2Net Learning, Inc.

Copyright © Ed2Net Learning, Inc. Graphing Functions EXAMPLE 1 Graphing a Function To graph the function y = 15x mentioned before, follow the steps below. Input, x Output, y 1 15 1 Make an input-output table for the function y = 15x. 2 30 3 45 2 Write the input and output values as ordered pairs: (input, output). (0, 0), (1, 15), (2, 30), (3, 45) Copyright © Ed2Net Learning, Inc.

Copyright © Ed2Net Learning, Inc. Graphing Functions EXAMPLE 1 Graphing a Function To graph the function y = 15x mentioned before, follow the steps below. Input, x Output, y 1 15 3 Graph the ordered pairs. Notice that the points lie along a straight line. 1 Make an input-output table for the function y = 15x. 2 30 3 45 If you chose other input values for the table, the points you would graph would also lie along the same line. • 2 Write the input and output values as ordered pairs: (input, output ) • 4 Draw a line through the points. (0, 0), (1, 15), (2, 30), (3, 45) (1.5, 22.5) is also on the line, because 15(1.5) = 22.5. 22.5 That line represents the complete graph of the function y = 15x. • • Copyright © Ed2Net Learning, Inc. 1.5

Representing Functions Graphing Functions NOTE BOOK Representing Functions There are many ways to represent the same function. Words A number is the sum of another number and one. Algebra y = x + 1 Ordered Pairs (–2, –1), (–1, 0), (0, 1), (1, 2), (2, 3) Input-Output Table Graph Input, x Output, y –2 –1 1 2 Copyright © Ed2Net Learning, Inc.

Copyright © Ed2Net Learning, Inc. Graphing Functions Types of Functions A linear function is a function whose graph is a straight line. Not all functions are linear functions. Copyright © Ed2Net Learning, Inc.

Copyright © Ed2Net Learning, Inc. Graphing Functions Types of Functions A linear function is a function whose graph is a straight line. Not all functions are linear functions. EXAMPLE 2 Identifying Linear Functions Tell whether the function is linear or not linear. Explain. The function is linear, because the graph is a straight line. Copyright © Ed2Net Learning, Inc.

Copyright © Ed2Net Learning, Inc. Graphing Functions Types of Functions A linear function is a function whose graph is a straight line. Not all functions are linear functions. EXAMPLE 2 Identifying Linear Functions Tell whether the function is linear or not linear. Explain. The function is linear, because the graph is a straight line. The function is not linear, because the graph is a not straight line. Copyright © Ed2Net Learning, Inc.

Copyright © Ed2Net Learning, Inc. Graphing Functions EXAMPLE 3 Looking for a Pattern Pools The graph shows the time it takes to fill a wading pool to various depths. Predict how long it takes to fill the pool to a depth of 15 inches. SOLUTION 1 Write some ordered pairs from the graph. (2, 6), (4, 12), (6, 18), (8, 24) 2 Write a function rule. where d is the depth in inches and t is the time in minutes. t = 3d, 3 Evaluate the function when d = 15. t = 3(15) = 45 ANSWER The water will be 15 inches deep in about 45 minutes. Copyright © Ed2Net Learning, Inc.

Copyright © Ed2Net Learning, Inc. Assessment 23 + y = 48 3/5h = 75 3x + 4 = 5 12(d + 5) = 72 + 9d A number more than 27 Copyright © Ed2Net Learning, Inc.

Copyright © Ed2Net Learning, Inc. Assessment 6. Twice Julian’s age 7 + z ≤ 13 8. 96 > 4k 9. 2(w - 10) ≥ -8 10. Check if the inequality is true for m = 3 42 ≤ 4(2n + 4) Copyright © Ed2Net Learning, Inc.

Copyright © Ed2Net Learning, Inc. You had a Great Lesson Today! Well Done! Be sure to keep practicing Copyright © Ed2Net Learning, Inc.