Algebraic Equations and Expressions

Ever wonder what makes equations different from expressions? It's simple - equations have equals signs and can be solved, while expressions just need to be evaluated. Remember to always follow PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction) when working with either.

When solving equations, follow these key steps:

1. Clear any fractions or decimals first
2. Simplify each side (focus on moving variables)
3. Add or subtract terms to isolate variables
4. Multiply or divide as needed (avoid dividing by fractions)

For example, solving 53 = 3(y-2) - 2(3y-1) requires distributing first: 53 = 3y-6-6y+2, then simplifying to 53 = -3y-4. Adding 4 to both sides gives 57 = -3y, which means y = -57/-3 = 19.

Pro Tip: Always check your answers by substituting them back into the original equation to verify you've found the correct solution!

Solving Absolute Value Equations

Absolute value equations might look intimidating, but they're actually straightforward once you understand the technique. Remember that absolute value represents distance from zero, so it's always positive or zero.

The key strategy is to get the absolute value expression by itself first. Then, since |x| = 5 means x = 5 OR x = -5, you'll need to write two separate equations to solve. For example, with |y+3| = 8, you'd write:

• y + 3 = 8 (which gives y = 5)
• y + 3 = -8 (which gives y = -11)

Be careful with negative signs outside the absolute value. In -2|6-4t| = -16, first isolate the absolute value by dividing both sides by -2, giving you |6-4t| = 8. Then solve both possibilities.

Remember: When your absolute value equals a negative number $like |x| = -5$, there's no solution because absolute value can never be negative!

Solving Inequalities

Inequalities are like equations with attitude - they tell you when values are greater than, less than, or equal to each other. The most important rule to remember is that when you multiply or divide by a negative number, you must flip the inequality symbol.

To solve an inequality like 4y - 3 < 5y + 2, start by getting variables on one side: 4y - 5y < 2 + 3, which simplifies to -y < 5. Multiply both sides by -1 (and flip the symbol!) to get y > -5.

For fractions in inequalities, multiply both sides by the LCD (least common denominator) to clear the fractions. In the example 3-2x > (x+5)/3, multiply everything by 3 to get 9-6x > x+5, then solve as normal.

Visual Tip: Always draw your solution on a number line to double-check your work. The direction of your arrow should match your inequality symbol!

Compound and Absolute Value Inequalities

Compound inequalities involve two conditions that must be satisfied at the same time. When you see "AND" conditions like -4 ≤ x < 3, this means x must be in between both values (greater than or equal to -4 AND less than 3).

"OR" conditions are different - they're satisfied when either condition is true. For example, x ≥ 5 OR x ≤ -3 means x can be either greater than/equal to 5 OR less than/equal to -3. On a number line, OR graphs typically point in opposite directions.

Absolute value inequalities follow special patterns:

• |ax + b| < c becomes -c < ax + b < c (values between -c and c)
• |ax + b| ≥ c becomes ax + b ≥ c OR ax + b ≤ -c (values outside the range)

For example, |2x - 1| ≤ 9 means -9 ≤ 2x - 1 ≤ 9, which becomes -4 ≤ x ≤ 5 after solving.

Quick Check: For absolute value inequalities, remember that |x| < c creates an interval, while |x| > c creates two separate regions on the number line.

Relations and Functions

Functions are special relationships where each x-value has exactly one y-value. Think of them as machines that take an input and produce exactly one output. Relations are more general - they're just sets of ordered pairs.

The domain is all possible x-values for your function, while the range contains all possible y-values. Functions can be:

• Discrete: Defined only for certain specific values (shown as dots on a graph)
• Continuous: Defined for all values within an interval (shown as unbroken lines)

How can you tell if a relation is a function? Use the vertical line test! If any vertical line crosses your graph more than once, it's not a function because that would mean one x-value has multiple y-values.

When evaluating a function like f$x$ = x² - 3 at x = -2, simply substitute: f$-2$ = $-2$² - 3 = 4 - 3 = 1. This gives you the ordered pair $-2, 1$.

Function Check: If you see the same x-value paired with different y-values in a relation, it automatically fails the function test - no graphing required!

Linear Equations and Lines

Linear equations create straight lines on a graph and can be written in different forms. The most common is slope-intercept form $y = mx + b$, where m represents the slope (rate of change) and b is the y-intercept (where the line crosses the y-axis).

To find the slope between two points, use the formula m = $y₂-y₁$/$x₂-x₁$. Be careful with vertical lines - their slope is undefined because the denominator would be zero! These are written as x = (some number).

When working with lines, remember:

• Parallel lines have the same slope
• Perpendicular lines have slopes that are negative reciprocals of each other (like 3 and -1/3)

To write an equation using point-slope form, use y - y₁ = m$x - x₁$, where (x₁, y₁) is a point on the line and m is the slope. For example, with point $5, -2$ and slope 3, you'd write y + 2 = 3$x - 5$, which simplifies to y = 3x - 17.

Application Alert: In real-world problems, the slope often represents the rate of change (like cost per hour), while the y-intercept represents the starting value (like an initial fee).