Unit 2: Solving Equations & Inequalities

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1 Unit : Solving Equations & Inequalities BASIC THINGS TO REMEMBER: 1. Always simplify both sides of the equation before you try to solve.. Keep it balanced! Whatever you do to one side of the equation, you do to the other. 3. In order to undo an operation, you perform the opposite operation. The opposite of addition is The opposite of subtraction is The opposite of multiplication is The opposite of division is 4. Every equation will end with a variable equal to a number.. To check an answer, you substitute it back into the problem. If you come up with a true statement then it is the solution. EX: Is x = a solution to the equation -x + 6 = -? -() + 6 = -4. So x= is not a solution. 6. The two things that help the most when you are learning to solve equations are: o o keep your work neat and organized do one step at a time, showing your work 7. Words that tell you to solve: evaluate solve find the value of simplify distribute and solve perform the indicated operations Solving Equations: Addition and Subtraction x 4 = 3 y = 4y y -4y x = 7 y = -1 NOTE: Before you even start Separate the two sides with a line! We can solve all of the equations below using only addition and/or subtraction. What s the opposite of +3? What s the opposite of -1? 1. x 3 1. y

2 3. 4 m t. ( 3) 1 Re-write so that there is only one sign! m y ( 4) x x 8. 4 x x Solving Equations: Multiplication & Division REMEMBER To solve an equation with multiplication, you by the number that is with the variable. To solve an equation with division, you by the number that is with the variable. Multiplying by the reciprocal is the same as. If a fraction is negative, you want to multiply by a number. Examples: y x 130 How is connected to x? 1 3 Move the by dividing. How is 3 connected to y? Move the 3 by multiplying. Solve each of the following equations using only multiplication or division. 1. 1x y a b 10

3 . 1 m c 6. 9 There are 3 ways to deal with fractions: 1) multiply by the reciprocal ) divide by the fraction 3) multiply by the bottom number then divide by the top number multiply divide two steps x 1 x 1 x z 4 6z 4 6z Solving Equations: Steps 1 st : the terms without numbers to get the variables on one side and the numbers on the other. nd : the term with the variable to separate the coefficient from the variable. Then we have only x = 8 left. We already know how to do that! The 1 is farthest from the x, so we move that first. Practice Problems: x. 3n 11

4 3. n n. m n Don t forget the sign belongs to the term AFTER it! x 8.Two consecutive odd integers have a sum of 1. Write the equation that shows this, then solve to find the larger of the numbers. Variables on Both Sides When we have variables on both sides of the equation, we 1. Move the variable to combine it with the variable on the other side by adding or subtracting it.. Add or subtract the number term to move it away from the variable. 3. Divide by the number beside the variable Practice Problems:

5 . 6. Solving Equations Multiple Steps 1. Distribute. 3x 6 1 x 13. Combine like terms on each side. Remember you only do 3x x 11 the opposite when you cross the line! 3. Add or subtract smaller variable x Add or subtract number to move it away from the variable.. Divide by the number with the variable. x 6 6. Check your answer! x 3 Practice Problems:

6 . Special Cases This means that the big messy fraction is the only thing on one side. Multiplying First The only time you can multiply first is when your problem has one side of the equation that looks like a big fraction. Variables Disappear If you know you did everything correctly and all of your variables disappear If you end with the same number on both sides, you write identity. If you end with a different number on each side, you write no solution Solving & Graphing Inequalities Solving: You solve an inequality ALMOST exactly the same as an equation, the one difference is If you multiply or divide by a NEGATIVE NUMBER, you switch the direction of the sign. Graphing: An inequality will often have many values that make it true, so the graph will be a ray. That is, one end is a point, the other is an arrow. 1. plot the number you get when you solve on a number line. decide whether or not to color it in 3. Pick an easy number on either side of the point and plug them into your answer ( > x; use 4 & 6) 4. Draw and arrow going the direction that makes it true. 14

7 greater than less than not equal to Sign Point is greater than or equal to less than or equal to equal to Sign Point is EX: To graph 1 > x, pick a number on either side of 1 (0 and are easy) 1 > 0 is true, so draw your arrow this way 1 > is not a true statement Practice Problems Solve each inequality and graph the solution

8 Ratios & Proportions A ratio is a comparison between two or more numbers in lowest terms. Ex: if there are 1 boys and 10 girls in a class, there is a boy-to-girl ratio of :. (Most of the time, you will see this as a fraction, so it would be ) A proportion is when two specific ratios are equal to each other. (A fraction equals a fraction) The ONLY time you are allowed to cross-multiply is when you have a proportion!!! Practice Problems Solve each proportion by cross multiplying. 1.. Start by writing two equations with multiplication on both sides. Don t forget to use parenthesis if you see a + or in part of the fraction! It helps to draw a picture!!. The scale drawing of a rectangular swimming pool has a length of 8 in. and a width of 4. in. If the actual pool has a length of 4 ft., what is the width? 16