The Full Guide

FITE Notes: Algebra I

Everything you need to pass Algebra I and stay on the field.

From Coach Fite

Listen. The classroom is part of the game now. If your grades drop, you don't play, period, and I didn't get walked through this stuff when I needed it, so I'm walking you through it. Algebra is just rules you do the same way every time, like a play you run til it's automatic. Read these notes, do the steps, pass the test, stay eligible. That's a win on Friday and a win on the report card. Let's work.

1

Variables, Expressions & Order of Operations

A variable is just a letter that stands in for a number you don't know yet, like x or y. An expression is numbers and variables combined with operations, like 3x + 5. It has no equals sign, so you don't solve it, you simplify it.

The order of operations (PEMDAS)

  • ●P - Parentheses (and other grouping) first
  • ●E - Exponents next
  • ●MD - Multiply and Divide, left to right
  • ●AS - Add and Subtract, left to right

REMEMBER

Multiply/Divide are the same rank, and Add/Subtract are the same rank. When they tie, go left to right. Don't always multiply before you divide.

  1. 1Simplify 2 + 3 * (4 + 1)^2
  2. 2Parentheses: 4 + 1 = 5, so 2 + 3 * 5^2
  3. 3Exponent: 5^2 = 25, so 2 + 3 * 25
  4. 4Multiply: 3 * 25 = 75, so 2 + 75
  5. 5Add: 2 + 75 = 77

Combining like terms

Like terms have the exact same variable part. 3x and 5x are like terms; 3x and 3x^2 are not. To combine, add or subtract the number in front (the coefficient) and keep the variable the same. Example: 4x + 7 + 2x - 3 becomes 6x + 4. You can only add x to x, and x^2 to x^2, and a plain number only combines with a plain number.

2

Solving One- and Two-Step Equations

An equation has an equals sign, and your job is to get the variable alone on one side. The golden rule: whatever you do to one side, do to the other side too. That keeps it balanced.

Undo with the opposite

  • ●Addition is undone by subtraction
  • ●Subtraction is undone by addition
  • ●Multiplication is undone by division
  • ●Division is undone by multiplication
  1. 1Solve 2x + 5 = 13
  2. 2Undo the +5 first: subtract 5 from both sides, giving 2x = 8
  3. 3Undo the times 2: divide both sides by 2, giving x = 4
  4. 4Check: 2(4) + 5 = 8 + 5 = 13, correct

REMEMBER

In a two-step equation, undo addition or subtraction FIRST, then undo multiplication or division. It's the reverse of PEMDAS.

Always check by plugging your answer back into the original equation. If both sides match, you got it.

3

Inequalities & The Flip Rule

Inequalities use the signs less-than, greater-than, less-than-or-equal, and greater-than-or-equal instead of an equals sign. You solve them almost exactly like equations, doing the same thing to both sides.

REMEMBER

If you multiply or divide both sides by a NEGATIVE number, flip the inequality sign. Less-than becomes greater-than, and so on. This is the one rule everyone forgets.

  1. 1Solve -2x + 1 less-than 9
  2. 2Subtract 1 from both sides: -2x less-than 8
  3. 3Divide both sides by -2 AND flip the sign: x greater-than -4

Graphing on a number line

  • ●Use an OPEN circle for less-than or greater-than (the number is not included)
  • ●Use a CLOSED (filled) circle for less-than-or-equal or greater-than-or-equal (the number is included)
  • ●Shade toward the numbers that make it true: right for greater-than, left for less-than

ON THE TEST

Open circle means the dot is hollow, the endpoint is not part of the answer. Filled circle means it counts.

4

Graphing Lines & Slope-Intercept Form

The coordinate plane has a horizontal x-axis and a vertical y-axis. A point is written (x, y): go x to the right or left, then y up or down. The middle, where they cross, is the origin (0, 0).

Slope

Slope measures how steep a line is. It is rise over run, meaning change in y divided by change in x. A positive slope goes up to the right, a negative slope goes down to the right. Slope of 2 means up 2, over 1.

Slope-intercept form: y = mx + b

  • ●m is the slope (steepness and direction)
  • ●b is the y-intercept, where the line crosses the y-axis
  • ●Example: in y = 3x + 2, slope is 3 and the line crosses the y-axis at (0, 2)
  1. 1Graph y = 2x - 1
  2. 2Start at the y-intercept b = -1, so plot the point (0, -1)
  3. 3Use slope m = 2 (up 2, right 1) to get the next point (1, 1)
  4. 4Draw a straight line through both points

ON THE TEST

Get the equation into y = mx + b form first. Then b tells you where to start, and m tells you how to step to the next point.

5

Writing Equations & Slope From Two Points

Sometimes you're given two points and asked for the line through them. First find the slope, then find the y-intercept.

Slope from two points

Slope m equals (y2 - y1) divided by (x2 - x1). Subtract the y's on top and the x's on the bottom, in the same order.

  1. 1Find the line through (1, 3) and (3, 7)
  2. 2Slope m = (7 - 3) / (3 - 1) = 4 / 2 = 2
  3. 3Plug a point and the slope into y = mx + b: 3 = 2(1) + b
  4. 4Solve for b: 3 = 2 + b, so b = 1
  5. 5The equation is y = 2x + 1

REMEMBER

Keep the order consistent. If the first y is on top, the first x must be on the bottom. Mixing the order flips the sign of your slope.

A horizontal line has slope 0 (no rise). A vertical line has an undefined slope (no run, you can't divide by zero).

6

Systems of Equations

A system is two equations with two variables. The solution is the (x, y) point that makes BOTH true at once, which is where the two lines cross. There are two main ways to solve.

Substitution

  1. 1Solve y = x + 1 and 2x + y = 7
  2. 2Substitute x + 1 in for y: 2x + (x + 1) = 7
  3. 3Combine and solve: 3x + 1 = 7, so 3x = 6, x = 2
  4. 4Plug back in: y = 2 + 1 = 3. Solution is (2, 3)

Elimination

  1. 1Solve x + y = 6 and x - y = 2
  2. 2Add the two equations to cancel y: 2x = 8, so x = 4
  3. 3Plug into either equation: 4 + y = 6, so y = 2. Solution is (4, 2)

ON THE TEST

Use substitution when one variable is already alone (like y = ...). Use elimination when adding or subtracting will cancel a variable. Both should give the same answer.

7

Exponents & Polynomials

An exponent tells you how many times to multiply a base by itself. In x^3, x is the base and 3 is the exponent, so x^3 = x times x times x.

Exponent rules

  • ●Multiply same bases: add the exponents. x^2 * x^3 = x^5
  • ●Divide same bases: subtract the exponents. x^5 / x^2 = x^3
  • ●Power of a power: multiply the exponents. (x^2)^3 = x^6
  • ●Anything to the 0 power is 1. x^0 = 1

Polynomials

A polynomial is terms added or subtracted, like 3x^2 + 2x - 5. To ADD them, just combine like terms. Example: (3x^2 + 2x) + (x^2 - 5x) = 4x^2 - 3x.

To MULTIPLY two binomials, use FOIL: First, Outer, Inner, Last. Example: (x + 2)(x + 3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6.

REMEMBER

When you MULTIPLY same bases you ADD exponents. Don't multiply the exponents unless it's a power raised to a power.

8

Factoring & Intro to Quadratics

Factoring is the reverse of multiplying. You take a polynomial like x^2 + 5x + 6 and break it back into two binomials that multiply to give it.

  1. 1Factor x^2 + 5x + 6
  2. 2Find two numbers that MULTIPLY to 6 and ADD to 5
  3. 3Those numbers are 2 and 3 (2 times 3 = 6, 2 + 3 = 5)
  4. 4Write it as (x + 2)(x + 3)

The zero-product idea

If two things multiply to zero, then at least one of them must be zero. So if (x + 2)(x + 3) = 0, then x + 2 = 0 or x + 3 = 0, giving x = -2 or x = -3. Those are the solutions of the quadratic.

ON THE TEST

Set the equation equal to 0 first, then factor, then set each factor to 0 and solve. A quadratic usually has two answers.

A quadratic equation (one with an x^2 term) graphs as a U-shaped curve called a parabola. It opens up if the x^2 term is positive and opens down if it's negative. The points where the parabola crosses the x-axis are the solutions you found by factoring.

Coach Fite

Factoring trips people up because it feels like guessing. It's not. Find the two numbers that multiply and add right, every time. Run it til it's automatic, same as footwork. Pass this unit and you've basically finished the year.

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