Comprehensive Algebra I Study Guide and Reference

Fundamental Rules for Integers and Equations

Integer Rules for Addition
- If the signs of the numbers are the same: Add the numbers together and keep the existing sign.
- If the signs of the numbers are different: Subtract the numbers and keep the sign of the number with the largest absolute value.
Integer Rules for Subtraction
- Rule: Add the opposite.
- Method: Keep—Change—Change
  - Keep: The first number remains the same.
  - Change: Change the subtraction symbol to an addition symbol.
  - Change: Change the sign of the second number to its opposite sign.
Integer Rules for Multiplication and Division
- If the signs are the same (both positive or both negative), the result is positive.
- If the signs are different (one positive and one negative), the result is negative.
The Golden Rule for Solving Equations
- "Whatever You Do To One Side of the Equation, You Must Do to the Other Side!"
Combining Like Terms
- Like terms are defined as two or more terms that contain the same variable.
- Examples of Like Terms:
  - $3x, 8x, 9x$
  - $2y, 9y, 10y$
- Non-examples: $3x, 3y$ are NOT like terms because they do NOT possess the same variable.
Distributive Property Examples
- $3(x+5) = 3x + 15$ : Multiply the $3$ by both $x$ and $5$ .
- $-2(y - 5) = -2y + 10$ : Multiply $-2$ by both $y$ and $-5$ .
- $5(2x - 6) = 10x - 30$ : Multiply $5$ by both $2x$ and $-6$ .

Seven-Step Study Guide for Solving Equations

Check for Fractions: Does your equation have fractions?
- If Yes, multiply every term on both sides of the equation by the denominator.
- If No, proceed to Step 2.
Check for Distributive Property: Does the equation involve the distributive property? (Are there parentheses?)
- If Yes, rewrite the equation using the distributive property.
- If No, proceed to Step 3.
Check for Like Terms: Are there like terms on either side of the equation?
- If Yes, rewrite the equation with the like terms placed together, then combine them. Note: Always take the sign located in front of each term.
- If No, proceed to Step 4.
Check for Variables on Both Sides: Are variables present on both sides of the equation?
- If Yes, add or subtract the terms to consolidate all variables on one side and all constants on the other side. Then skip to Step 6.
- If No, proceed to Step 5.
Identify the Two-Step Equation: At this point, the equation should be a basic two-step equation. If it is not, recheck previous steps.
- Use Addition or Subtraction to remove constants from the variable side. (Apply the Golden Rule).
Isolate the Variable: Use multiplication or division to remove any coefficients from the side of the variable. (Apply the Golden Rule).
Verify the Result: Check the answer using substitution into the original equation.

Graphing Linear Equations

Slope $(\text{m})$
- Formula: $\text{Slope} = \frac{\text{rise}}{\text{run}}$
- Calculation Method:
  - Choose two specific points on the line.
  - Count the rise: How far up or down is required to reach the next point? This value is the numerator.
  - Count the run: How far left or right is required to reach the next point? This value is the denominator.
  - Write the slope as a fraction (e.g., $\text{Slope} = \frac{3}{5}$ ).
- Graph Reading Rules:
  - Read graphs from left to right.
  - If the line is falling, the slope is negative.
  - If the line is rising, the slope is positive.
  - When counting: Down or Left is negative. Up or Right is positive.
Slope-Intercept Form
- Equation: $y = mx + b$
- $m$ represents the Slope.
- $b$ represents the Y-intercept.
Graphing Steps Using Slope-Intercept Form
1. Identify the slope and y-intercept (e.g., in $y = 3x - 2$ , the slope is $3$ and the y-intercept is $-2$ ).
2. Plot the y-intercept on the y-axis.
3. From the y-intercept point, count the rise and run determined by the slope to locate the second point.
4. Draw a continuous line through both points.

Writing Equations of Lines

Basic Writing with Slope and Y-intercept
- If Slope $= 3$ and Y-intercept $= -4$ , the equation is $y = 3x - 4$ .
Writing Equations Given Slope and One Point
- You are provided with $m, x$ , and $y$ .
- Step 1: Substitute $m, x$ , and $y$ into $y = mx + b$ and solve to find $b$ .
- Step 2: Use the given $m$ and the calculated $b$ to write the final equation in slope-intercept form.
- Example: Slope $= 2$ , Point $(3, 1)$ .
  - $m = 2, x = 3, y = 1$
  - $1 = 2(3) + b$
  - $1 = 6 + b$
  - $1 - 6 = 6 - 6 + b$
  - $-5 = b$
  - Equation: $y = 2x - 5$
Standard Form
- Equation: $Ax + By = C$
- Requirements: $A, B$ , and $C$ must be integers. Importantly, $A$ must be a positive integer.
- Example Conversion:
  - $-3x + 2y = 9$ (Incorrect because $A$ is negative).
  - Multiply all terms by $-1$ to get: $3x - 2y = -9$ (Correct).
Writing Equations Given Two Points
- Step 1: Find the slope $(m)$ using the formula: $\frac{y_2 - y_1}{x_2 - x_1}$ .
- Step 2: Use the calculated slope and one of the two given points to find the y-intercept $(b)$ .
- Step 3: Write the final equation.
- Example: Points $(1, 6)$ and $(3, -4)$ .
  - Step 1: Slope $m = \frac{-4 - 6}{3 - 1} = \frac{-10}{2} = -5$ .
  - Step 2: Apply to $y = mx + b$ using point $(1, 6)$ .
    - $6 = -5(1) + b$
    - $6 = -5 + b$
    - $6 + 5 = -5 + 5 + b$
    - $11 = b$
  - Step 3: Equation: $y = -5x + 11$ .

Systems of Equations

Definition: Two linear equations together form a system. The solution is the point of intersection $(x, y)$ that satisfies both equations.
Solution Types:
- Intersection: One unique solution.
- Parallel Lines: No solution (lines never intersect).
- Coinciding Lines: Infinite solutions (one line lies directly on top of the other).
Substitution Method
- Solve: $x - 2y = -10$ and $y = 3x$ .
1. Substitute $3x$ for $y$ in the first equation: $x - 2(3x) = -10$ .
2. Simplify: $x - 6x = -10 \rightarrow -5x = -10$ .
3. Solve for $x$ : $\frac{-5x}{-5} = \frac{-10}{-5} \rightarrow x = 2$ .
4. Substitute $x = 2$ back into $y = 3x$ : $y = 3(2) = 6$ .
5. Solution: $(2, 6)$ .
Linear Combinations (Addition Method)
- Solve: $3x + 2y = 10$ and $2x + 5y = 3$ .
1. Create opposite terms for one variable. Multiply the first by $-2$ and second by $3$ to target $x$ :
  - $-2(3x + 2y = 10) \rightarrow -6x - 4y = -20$
  - $3(2x + 5y = 3) \rightarrow 6x + 15y = 9$
2. Add the equations: $(-6x + 6x) + (-4y + 15y) = -20 + 9 \rightarrow 11y = -11$ .
3. Solve for $y$ : $y = -1$ .
4. Substitute $y = -1$ into an original equation: $2x + 5(-1) = 3 \rightarrow 2x - 5 = 3 \rightarrow 2x = 8 \rightarrow x = 4$ .
5. Solution: $(4, -1)$ .

Inequalities

Inequality Symbols
- $<$ : Less Than
- $\leq$ : Less Than OR Equal To
- $>$ : Greater Than
- $\geq$ : Greater Than or Equal To
The Negative Multiplier/Divisor Rule
- Whenever you multiply or divide an inequality by a negative number, you MUST reverse the inequality sign.
- Example: $-3x < 9$
  - Divide by $-3$ : $\frac{-3x}{-3} > \frac{9}{-3}$
  - Result: $x > -3$
Graphing Inequalities in Two Variables
- Example: $y > -\frac{1}{2}x + 1$
1. Graph the boundary line $y = -\frac{1}{2}x + 1$ . Use a dotted line because the symbol is $>$ , meaning points on the line are not solutions.
2. Use a test point such as $(0, 0)$ . Substitute: $0 > -\frac{1}{2}(0) + 1 \rightarrow 0 > 1$ . This is False.
3. Shade the side of the line that does not contain the point $(0, 0)$ .
Systems of Inequalities
- Graph both inequalities on the same coordinate plane. The solution set is ONLY the area shaded by both inequalities (often appearing as an overlapping orange section).

Functions and Quadratics

Vertical Line Test
- Identifying Functions: A graph is a function if every vertical line drawn through it touches the graph exactly one time. If a vertical line touches more than once, it is not a function.
Function Notation
- $f(x) = 3x + 2$ (Translated: "f of x equals $3x + 2$ ")
- $g(x) = 3x - 1$ (Translated: "g of x equals $3x - 1$ ")
Quadratic Functions
- Identification: Contains a squared term (e.g., $x^2$ ).
- Geometric Shape: Results in a parabola when graphed.
- Parabola Direction:
  - If the leading coefficient is positive (e.g., $3x^2 + 2x - 5$ ), the parabola opens up.
  - If the leading coefficient is negative (e.g., $-2x^2 + 2x - 5$ ), the parabola opens down.
- Vertex Formula:
  - Given $f(x) = ax^2 + bx + c$
  - Formula for the x-coordinate of the vertex: $x = \frac{-b}{2a}$ (the opposite of $b$ divided by $2$ times $a$ ).

Exponents, Monomials, and Polynomials

Laws of Exponents
- Multiplying Powers with the Same Base: Add the exponents.
- Power of a Product Property: Apply the outer exponent to every factor inside the parentheses.
- Power of a Power Property: Multiply the exponents.
- Power of Quotient Property: Raise the numerator and the denominator to the power, then divide.
- Zero Exponents: Any non-zero base raised to the power of zero equals $1$ .
Operations with Polynomials
- Adding Polynomials: Only combine like terms.
- Subtracting Polynomials: Use the Keep-Change-Change rule to change subtraction to addition and adjust the signs of the second polynomial.
- Multiplying Polynomials: Use the laws of exponents. For binomials, use the FOIL method (First, Outer, Inner, Last).
Factoring Techniques
- Greatest Common Factor (GCF): Identifying the largest factor shared by all terms.
- Factoring Using GCF: Dividing each term by the GCF and writing the result as a product.
- Factoring by Grouping: Used for polynomials with four terms.
- Factoring Trinomials: Finding binomial factors that multiply to the original expression.
Quadratic Equations
- Definition: An equation where the highest exponent of the variable is $2$ .
- Methods of solving: Solving simple quadratics (taking square roots) or solving by factoring.