Chapter 3. linear equation and linear equalities in one variables
Open SentencesUnderstanding Linear Equations in One Variable Solving Linear Equations in One Variable Applying Equations in Everyday life The Graph of Solutions of Equations Understanding Inequalities Solving Linear Inequalities in One Variable The Graph of an Inequality Applying Inequalities
A sentence which contains one or more variables such that its truth (being true or false) can not be determined is called an open sentence.Example : x 7 15 A variable is a symbol that can be replaced with an arbitrary number.From the example, X is a variable. The subtitute for a variable which makes an open sentence a true sentence is called a solution.From the example, if x is replaced with 8, the sentence will be true.Thus, its solution is x 8
What’s a Linear Equation in One Variable ?A linear equation in one variable is anequation that can be written in the form ax + b = 0Where a 0 For example: 2x + 6 = 0
Solving EquationsTo solve an equation means to find allvalues that make the equation a truestatement. Such values are calledsolutions, and the set of all solutions iscalled the solution set.
Objectives:• * To solve linear equations using addition and subtraction• * To use linear equations to solve word problems involving real-world situations• Your goal is to isolate the variable and keep sides balanced • Watch integer signs and remember the rules • Fraction and decimal rules still apply
To solve equations like x 3 6 , you can use mentalmath.Ask yourself, “What number minus 3 is equivalent to 6?” x This strategy can work for easier problems, but we need a better plan so we can solve more difficult problems. In this strategy, we balance the sides of the equation as we solve for the variable. First, we must “undo” the minus 3. The inverse x 3 6 (opposite) of subtracting 3 is _________________ 3.x 3 3 6 3 But we have to be fair! If we are going to add 3 to one side of the equation, we MUST add 3 to x 9 the other side to BALANCE the equation.
• RULE: Subtract the same amount from each side of the equation to keep the equation balanced• Algebraic symbols: for any numbers a, b, and c, if a=b, then a-c= b-c
The process for solving equations using subtraction is very similar to solving equations using addition. How do you undo addition?Example: First, we must “undo” the plus 2. The inverse x 2 5 (opposite) of adding 2 is ______________ 2. x 2 -2 5 -2 But we have to be fair! If we are going to subtract 2 from one side of the equation, we MUST subtract 2 from the other side to BALANCE the x 3 equation. Ex1) x 7 9 Ex2) x 15 10
In some problems, you will see a negative sign in frontof the variable you are solving for.Example: x 12 What does this mean? x 12 x = -12 The negative sign means “the opposite of x,” meaning the opposite sign, positive or negative.To solve, we change the sign of the other side of the equation. 4 2-x = -5 x 2 7 z 6 ( x) 7 3 5
• Focus: How can equations be used to find how long it takes light to reach Earth?• What is the unknown quantity in the equation given in the example?• What variable represents the unknown quantity in the equation?• What do you need to accomplish to solve this equation?
• REMEMBER: Your goal is to isolate the variable!• Multiplication Property: Multiply each side of the equation by the same number to keep the equation balanced• Division Property: Divide both sides of the equation by the same number to keep the equation balanced
Objectives: To solve linear equations using multiplication and division and touse linear equations to solve word problems involving real-world situations To solve equations like x 2 , you will need to use multiplication. 4 x In this problem, the x is being divided by 4. To 2 solve for x, we will need to do the inverse of 4 dividing by 4, which is multiplying by 4. x **Don’t forget that you will need to do this to 4 42 BOTH sides of the equation to keep it balanced! 4 Ex) a a x 8 10 10 2 2
To solve equations like 4x 32 , we can use division. 4x 32 In this problem, the x is being multiplied by 4. To solve for x, we will need to do the inverse of 4x 32 multiplying by 4, which is DIVIDING by 4. **Don’t forget that you will need to do this to 4 4 BOTH sides of the equation to keep it balanced! x 8
Objectives: To use 2 or more steps to solve a linear equation and to usemulti-step equations to solve word problems You will be asked to solve problems like 3x 1to 7 which require more than one step solve.Guidelines for Solving Multi-Step Equations:1) Simplify both sides if necessary distribute & combine like terms2) To solve you must undo the order of operations BACKWARDS! Start with Undo add or subtract Then Undo multiplication and division 3x 1 7 There is nothing to simplify What goes first? 1 1 Simplify to get a new equation. 3x 6 What goes next? 3 3 How can you check you answer? Do I expect you to show work on x 2 assignments, quizzes and tests?
Lots of steps! Where should you start?? 3( x 2) 4 x 12 1 distribute 3( x 2) 4 x 12 1 simplify Combine like terms 3x 6 4x 13 7x 6 13 6 6 7x 7 7 7 x 1
1st: Variables to one side How do you decide who tomove?2nd: Constants to the other side Who must you move? 1) Which side has the smaller 7x + 19 = -2x + 55 coefficient?+2x +2x 2) Add 2x to both sides. 3) Simplify. 9x + 19 = 55 - 19 -19 4) Subtract 19 from both sides. 9x = 36 5) Simplify. 9x 36 6) Divide both sides by 9. 9 9 7) Simplify. x 4 Can you check your answer? How?
The Graph of Solutions of Equations 11x 2 5(x 2) 11x 2 5x 10 -5x -5x 6x 2 10 6x 12 -2 -2 6 6 6x 12 x 2The graph is -5 -4 -3 -2 -1 0 1 2 3 4 5
Applying equations in everyday life To solve problems encountered in daily life, especially the word problems, the following steps can help make the solutin finding easier. – If you feel the need for a diagram or a sketch, e.g., for a problem related to geometry, make a diagram or a sketch based on the story. – Translating the story into a mathematical sentence or mathematical model in the form the equations. – Solve the equations.
Several record temperature changes have taken place in Spearfish, South Dakota. On January 22, 1943, the temperature in Spearfish fell from 54 degrees Fahrenheit at 9:00 am to -4 degrees Fahrenheit at 9:27 am. By how many degrees did the temperature fall? You started with some money in your pocket. All you spent was $4.65 on lunch. You ended up with $7.39 in your pocket. Write an equation to find out how much money you started with.Can you solve one – step problems involving addition and subtr
1. First temperature = 54 degrees Fahrenheit, Last temperature = (-4) degrees Fahrenheit, Let x = the fall of temperature. Write an equations to solve the problem, the equations is 54 x 4 54 4 x 58 x So, the fall of temperature is 58 degrees Fahrenheit.2. Let I started with X money, spent : $ 4,65 and end up : $ 7,39. Then, write an equations to find out the value of X, the equations is x 4, 65 7,39 x 7,39 4, 65 x 12, 04 So, I started with $ 12,04
Ex 1.) It takes 6 cans of chili to make one batch of George’s extra- special chili-cheese dip. How many cans does it take to make 3 and a half batches? Write an equation and solve it. Be sure to state what your variables represent. Ex 2.) Amy’s mom has been passing out cookies to the members of Amy’s soccer team. If each of the team members (including 15 players, 2 coaches, and a manager) receive 3 cookies with no extra cookies left, how many cookies did Amy’s mom bring? Write an equation an solve it.Can you solve equations involving multiplication and division?
1) 1 batch 6 cans 2) The number of team members : 3 ½ batch X cans 15+2+1=18 persons We can write the equations: 1 person 3 cookies 18 persons X 1 6 7 x We can write the equations : 2 1 3 7 x.1 6. 18 x 2 x 3.7 x.1 3.18 x 21 x 54 So, George need 21 cans of chili to make 3 and a half So, Amy’s mom bring 54 cookies batches.
Ex) The bill (parts and labor) for the repair of a car was $400. The cost of parts was $125, and the labor cost was $50 per hour. Write and solve and equation to find the number of hours of labor.Can you solve multi-step equations? How do youknow what operation to undo first?
The bill (parts and labor) = $ 400.Cost of parts = $ 125.Cost of labor per hour = $ 50.Let X is the number of hours of labor.The equation is 400 125 50 x 400 125 50 x 275 50 x 275 x 50 5,5 x So, the number of hours of labor = 5,5 hours.
A linier inequality is an open sentence which has the relation of ≥, >, ≤, or < and whose variable has an exponent of 1.A linier inequality which has one variable iscalled a linear inequality in one variable (LIOV).The general form of a LIOV: ax b...0 , where ... can be one of , , ,or
Solving an InequalitySolving a linear inequality in one variable is much like solving a linearequation in one variable. Isolate the variable on one side usinginverse operations. Solve using addition: x–3<5 Add the same number to EACH side. x 3 5 +3 +3 x<8
Solving Using SubtractionSubtract the same number from EACH side. x 6 10 -6 -6 x 4
Multiply each side by the same positive number. 1 (2) x 3 (2) 2 x 6
Divide each side by the same positive number. 3x 9 3 3 x 3
Solving by multiplication of a negative #Multiply each side by the same negative number and REVERSE theinequality symbol. Multiply by (-1). (-1) x 4 (-1) See the switch x 4
Solving by dividing by a negative #Divide each side by the same negative number and reversethe inequality symbol. 2x 6 -2 -2 x 3
When you multiply or divide each side ofan inequality by a negative number, youmust reverse the inequality symbol tomaintain a true statement.
Solutions….You can have a range of answers…… -5 -4 -3 -2 -1 0 1 2 3 4 5 All real numbers less than 2 X<2
Solutions continued… -5 -4 -3 -2 -1 0 1 2 3 4 5All real numbers greater than -2 x > -2
Solutions continued…. -5 -4 -3 -2 -1 0 1 2 3 4 5All real numbers less than or equal to 1 x 1
Solutions continued… -5 -4 -3 -2 -1 0 1 2 3 4 5 All real numbers greater than or equal to -3 x 3
Did you notice, Some of the dots were solid and some were open? x 2 -5 -4 -3 -2 -1 0 1 2 3 4 5 x 1 -5 -4 -3 -2 -1 0 1 2 3 4 5 Why do you think that is?If the symbol is > or < then dot is open because it can not be equal.If the symbol is or then the dot is solid, because it can be that point too.
Applying inequlities in everyday life To solve inequality problems in the form of word problems, translate the problems first into inequalities, and then solve them. If necessary, make a diagram or sketch to make the solution finding easier.
Problems• The length of a rectangle is 6 cm longer than its width and its perimeter is less than 40 cm. If the rectangle’s width is x, then form an inequality in x and solve it!
• Answer:Width = x cm, length = (x+6) cm.Perimeter = 2p + 2l. 2p 2l 40 2( x 6) 2x 40 x cm 2x 12 2x 40 4x 12 40 (x+6) cm 4x 40 12 4x 28 4 x 28 Since a length and a width cannot be negative, then its 4 4 solutions is x 7 0 x 7