Unit 3: Linear Equations and Inequalities Section 1: Solving One-Step Equations When solving an equation, use the opposite operation to eliminate anything on the same side of the equation as the variable The goal is to get the variable alone (to solve for the variable) When solving ANY type of equation, you MUST show work to earn full credit Always write the final answer as the variable equals the value (i.e. x = 5)

Solve. Ex1. x + 5 = 12 Ex2. m – 9 = -13 Ex3. 8x = 72 Ex4. -3n = 72 Ex5. Ex6. Sections of the book to read: 2-6, 3-2, 4-3

Section 2: Solving One-Step Inequalities Inequalities are solved just like equations with one exception: whenever you multiply or divide both sides by a negative number, you change the sense of the inequality (change the direction of the inequality sign) Know the inequality symbols: is greater than, is greater than or equal to Multiplication Property of Inequality: If x ay

It is math etiquette to write the answer with the variable on the left side of the inequality Solve. Ex1. x – 9 > -7 Ex2. 12 < 4 + m Ex3. -2n > 36 Ex4. 6w > -42 Ex5. Ex6.

When graphing the solution to an inequality on a number line, remember to write the variable on the right-hand end of the line Have at least 5 numbers on your number line Open circle for Closed circle for Graph on a number line Ex7. x 8 Section of the book to read: 1-2, 1-3, 2-8, 3-10

Section 3: Special Numbers in Equations You will not always get a simple numerical answer to an equation Ex1. Solve 0x = 53 There is no solution here because you cannot multiply 0 by any number to get anything other than 0 (also think of it as dividing by 0, which is impossible) Ex2. 0y = 0 This is because you can multiply 0 by anything and get 0

Ex3. 7n = 0 This is because the only number you can multiply 7 by to get 0 is 0 Ex4. -z = 12 This is due to the Multiplication Property of -1 Without showing work, you should quickly be able to identify the answer to these types of questions Sections of the book to read: 2-5, 2-7

Section 4: Solving Two-Step Equations You must eliminate both numbers on the same side as the variable Remove them in the reverse order from the order of operations Your work should show what you did for each step and what was the result for each step To keep the equation balanced, remember that whatever you do to one side of the equation, you must do to the other side as well

Solve Ex1. 3x – 8 = -17 Ex2. 4(m + 5) = 20 Ex3. Ex4. Ex5. Section of the book to read: 3-5

Section 5: Solving Two-Step Inequalities Remember to change the sense of the inequality whenever you multiply or divide by a negative number Solve. Ex1. 3(x – 4) > -15 Ex2. 8 – 2x < 22 Ex3.

Ex4. Solve and graph the solution on a number line Section of the book to read: 3-10

Section 6: Solving Multi-Step Equations The first step is to simplify each side completely (this may already be done for you) The second step is to get the variables on the same side (preferably the left side) The third step is to get the constant terms together on the opposite side (the right side) The final step is to divide to get the variable alone

Solve Ex1. 5x + 3 – 2x = 8 + 7x + 1 Ex2. 4(2x + 1) – 5 = 3x – 7 + 7x Ex3. 5(3x – 6) = 2(3x + 4) – 5 If the equation has fractions in it, typically it is simpler to first multiply through by the common denominator to eliminate all fractions Ex4. Ex5.

It might be helpful to multiply through by a power of 10 so as to eliminate all decimals (you need to move the decimal point enough places so that all decimals are cleared) Ex6..002x = x You can also multiply through to reduce large numbers Ex n – 2400 = n Section of the book to read: 5-3, 5-8, 5-9

Section 7: Solving Multi-Step Inequalities Use all of the same rules as you used in solving equations, just remember to change the sense of the inequality if you multiply or divide by a negative number Write the variable on the left side of the solution You can still use the multiplying through technique

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. Ex4. 80, ,000m – 20,000 > -86,000m Sections of the book to read: 5-6 and 5-8

Section 8: Equivalent Formulas Equivalent formulas are equal, they have just been manipulated to look different Ex1. Solve 3x + y = 8 for y 3x + y = 8, y = 8 – 3x and y = -3x + 8 are all equivalent formulas, they are just written differently Ex2. Solve 4x – 6y = 12 for y Slope-intercept form for an equation of a line is y = mx + b

The m in slope-intercept form stands for slope Slope is typically written as an integer or as a fraction (rarely as a decimal) The b in slope-intercept form stands for the y-intercept Ex3. Write 7x + 5y = 9 in slope-intercept form Ax + By = C is standard form for an equation of a line A, B, and C must be integers (no fractions or decimals) and most books do NOT allow A to be negative

Ex4. Write in standard form A practical place where equivalent formulas are used is in converting temperatures In the US, we use the Fahrenheit scale Most of the rest of the world uses the Celsius or Centigrade scale To convert from Fahrenheit to Celsius, use the following formula Ex5. Solve the previous formula for F to find the equivalent formula for converting from Celsius to Fahrenheit

Ex6. Solve a = b + bc for c Sections of the book to read: 1-5, 5-7, 7-4, 7-8