Linear equations and Inequalities UNIT 3

Section 1 Solving One-Step Equations and Inequalities Use the opposite operation to isolate a variable Be sure to do the same thing to BOTH sides You MUST show work any time you solve an equation in any of my classes Solve. Ex1. x + 5 = 12Ex2. m – 9 = -13 Ex3. 8x = 72Ex4. Ex5. Solve an inequality just like an equation except if you multiply or divide by a negative number, switch the direction of the inequality sign (change the sense)

Inequality symbols: greater than, > greater than or equal to Multiplication Property of Inequality: If x ay It is a good idea to write the answer with the variable on the left Solve Ex6. x – 9 > -7Ex7. 12 < 4 + m Ex8. -2n > 36Ex9. When graphing inequalities, write the variable on the right side of the number line & show at least 5 numbers

Open circles mean that the number is NOT included in the answer Used for < and > Closed circles mean that the number is included in the answer Used for < and > Graph on a number line Ex10. x < -4Ex11. m > 8 Sections of the book to read: 1-2, 1-3, 2-8, and 3-10

Section 2: Solving Two-Step Equations Eliminate both numbers on the same side as the variable in the reverse order from the order of operations Your work must show what you did for each step as well as the result of that work Solve. Ex1. 3x – 8 = -17Ex2. 4(m + 5) = 20 Ex3. Ex4. Ex5. Section of the book to read: 3-5

Section 3: Solving Two-Step Inequalities Solve just like an equation except you must change the direction of the inequality sign if you multiply or divide by a negative number Solve Ex1. 3(x – 4) > -15Ex2. 8 – 2x < 22 Ex3. Ex4. Solve and graph the solution on a number line Section of the book to read: 3-10

Section 4: Solving Multi-Step Equations Begin by simplifying each side as much as possible Second, add or subtract to get the variable on one side Next, add or subtract to get the constant term on the other side Finally, divide to isolate the variable Solve Ex1. 5x + 3 – 2x = 8 + 7x + 1 Ex2. 4(2x + 1) – 5 = 3x – 7 + 7x Ex3. 5(3x – 6) = 2(3x + 4) – 5 To avoid working with fractions you can multiply through by the lowest common denominator first (eliminating the fractions but maintaining the equation)

Multiply through and solve Ex4.Ex5. You can also use multiplying through to eliminate decimals Multiply through by a power of 10 (the lowest power of 10 that will eliminate all of the decimals) Ex6. Multiply through and solve.002x = x Multiplying through can also work to make large numbers more manageable Ex7. Multiply through and solve 5200n – 2400 = n Sections of the book to read: 5-3, 5-8, and 5-9

Section 5: Solving Multi-Step Inequalities Solve just like an equation, except that you must change the direction of the inequality symbol if you multiply or divide by a negative number This also holds true if you multiply through by a negative Etiquette states that you write the answer with the variable on the left Solve. Write answers in fraction form where necessary Ex1. -5x x < 5(2x + 3) – x Ex2. 3(x – 8) – 4(2x + 1) > 4x + 7 Ex3. Sections of the book to read: 5-6 and 5-8

Section 6: Equivalent Formulas Equivalent formulas are equal to one another, one of them has just been manipulated to look different Ex1. Solve for y. 3x + y = 8 3x + y = 8 and y = 8 – 3x and y = -3x + 8 are all equivalent formulas Ex2. Solve for y. 4x – 6y = 12 Slope-intercept form of a line is y = mx + b m is the slope (usually a fraction) and b is the y-intercept Ex3. Write 7x + 5y = 9 in slope-intercept form Ax + By = C is standard form of a line A, B, and C must be integers and A must not be negative (although this book allows it)

Ex4. Write in standard form Equivalent formulas are used for things like converting temperatures to different scales To convert from Fahrenheit to Celsius, use the following formula: Ex5. Solve the previous formula for F (in order to find the formula to convert from Celsius to Fahrenheit Ex6. Solve for c. a = b + bc Sections from the book to read: 1-5, 5-7, 7-4, and 7-8

Section 7: Ratios and Proportions A ratio is a quotient of quantities with the SAME units Ratios can be written as fractions or decimals (never mixed numbers) Ex1. Find the ratio of boys to girls in this class Ex2. It takes Mr. Garcia.75 hours to get to work while it takes Mrs. Wong 12 minutes to get to work. Find the two ratios that compare their travel times. To find the percent of discount, divide the amount of the discount by the original price then convert to percent Ex3. A TV regularly sells for $595, but is on sale for $525. What is the percent of discount? To find the percent of tax, divide the amount of tax by the price then convert to percent

Ex4. The tax on a $79.95 item was $5.20. What is the percent of tax? If you know the ratios used in creating a mixture, you can use those to write an equation and solve Ex5. Instructions for making fruit punch cocktail call for 3 parts fruit punch and 1 part Sprite. How much of each ingredient is needed to make 20 gallons? A proportion is two ratios set equal to one another Cross-multiply to solve proportions Ex6. Solve Ex7. Car A traveled 360 miles on 14 gallons of gas. With the same driving conditions, how far could the car go on 20 gallons of gas?

Ex8. On a map, 1 inch = 600 miles. If two cities are inches apart on the map, how far apart are they in real life? Ex9. Solve Sections of the book to read: 6-3 and 6-8