Understanding Real Numbers

Real numbers are the numbers we use in everyday life. They include positive, negative, whole, and decimal numbers. But not all numbers are real! Imaginary numbers $like √-1$ and undefined expressions (like division by zero) aren't part of the real number family.

Real numbers can be organized into different categories. Counting numbers (1, 2, 3...) are the positive whole numbers we learn first. Integers include all whole numbers and their negatives (-2, -1, 0, 1, 2...). Rational numbers can be written as fractions of integers and have either terminating or repeating decimals.

Irrational numbers like π, √2, and e can't be written as simple fractions and have non-repeating, non-terminating decimals. Remember that real numbers can be visualized as an inclusive set: counting numbers are inside integers, which are inside the broader set of rational numbers, and all together with irrational numbers make up real numbers.

💡 When solving real-world problems, look for relationships between variables. For example, if W = number of weeks and D = number of days, you can write the equation D = 7W to convert between them.

Mathematical Modeling in Real Life

Mathematical models help us describe real-world situations using equations. Let's see how this works! When Joe makes $12 per hour, we can model his pay with the equation P = 12h, where P is pay and h is hours worked. This simple equation lets us solve various problems.

With this model, we can calculate that Joe earns $432 after working 36 hours. We can also determine that if Joe earned$60, he must have worked 5 hours. For any wage situation, we can generalize the model as P = wh, where w is the hourly wage.

Mathematical modeling is especially useful for rate problems. For instance, when calculating gas mileage (G), we divide miles traveled (m) by gallons used (l): G = m/l. If Joe drove 230 miles using 10.5 gallons, his gas mileage would be approximately 21.9 miles per gallon.

💡 The word "per" in a problem often indicates division or a rate. When you see "miles per gallon" or "dollars per hour," you're looking at a ratio that can be written as a fraction.

Real Number Properties

The real number line helps us visualize and compare numbers. When placing numbers like 1.325, 7/5 (which equals 1.4), √2 (approximately 1.41), and π (about 3.14) on the line, we can easily see their order: 1.325 < 7/5 < √2 < π.

Understanding decimal notation is crucial. A number like 3.179 means "3 and 179 thousandths" or 3 + 179/1000. For repeating decimals, we use a bar over the repeating part, such as 0.333... written as 0.3̅ or 1.2717171... written as 1.27̅1̅7̅.

Real numbers follow important properties that make algebra work. The commutative properties $a + b = b + a and ab = ba$ let us rearrange terms. The associative properties let us regroup terms, while the distributive property $a(b + c) = ab + ac$ allows us to multiply across addition.

💡 When working with absolute value, remember that the distance between two points A and B on the number line is |A - B|. This is useful for measuring how far apart two numbers are, regardless of which is larger.

Fractions and Algebraic Expressions

Working with algebraic fractions follows the same rules as regular fractions. To simplify expressions like $x+1$/y, look for ways to combine terms in the numerator: $x+1$/y + y/y = $x+y$/y.

When dividing fractions, remember to multiply by the reciprocal of the divisor. For example, $x+y$/y ÷ $x-y$/x = $x+y$/y · x/$x-y$. This gives us x$x+y$/y$x-y$, which can be expanded to $x²+xy$/$yx-y²$.

Factoring is a powerful tool for simplifying complex expressions. When multiplying fractions with polynomials, factoring first often makes the work much easier. For example, in the expression $x²+7x+12$/$x²+5x+2$ · $x²+5x+6$/$x²+6x+9$, factoring gives $x+3$$x+4$/$x+2$$x+1$ · $x+2$$x+3$/$x+3$$x+3$, which simplifies to $x+4$/$x+1$.

💡 When adding or subtracting fractions with different denominators, create a common denominator first. For example, 1/$x-1$ + 1/$x+3$ becomes $x+3$/$(x-1)(x+3)$ + $x-1$/$(x-1)(x+3)$, which simplifies to 2/$x+3$$x-1$.

Solving Equations

Solving equations is about isolating the variable. With quadratic equations $x² - 4 = 5$, add or subtract the same value from both sides to isolate the squared term $x² = 9$, then take the square root to find x = ±3. Remember to include both the positive and negative solutions!

When dealing with expressions like $x-4$² = 5, first take the square root of both sides to get x-4 = ±√5, then solve for x: x = 4 ± √5. For equations with fractional exponents, raise both sides to the reciprocal power—for x^(3/2) = 8, raise both sides to the power of 2/3 to get x = 4.

For equations with multiple terms with the same variable, group like terms before solving. In 2x + 7x = 2$-3x + 4$, expand the right side to -6x + 8, then combine all x terms: 2x + 7x + 6x = 8, which gives 15x = 8, so x = 8/15.

💡 When dealing with equations involving radicals, like √2x = √3x, isolate the radical terms on one side $√2x - √3x = 0$, then factor out the common terms: x(√2 - √3) = 0. This gives you two possible solutions: x = 0 or √2 = √3 (which is impossible).

Linear Equations and Slope

Linear equations connect points on a plane with a straight line. The slope-intercept form $y = mx + b$ is one of the most useful ways to write these equations, where m is the slope and b is the y-intercept.

To find the equation of a line through two points, first calculate the slope using the formula m = $y₂ - y₁$/$x₂ - x₁$. For points (1,2) and (3,7), the slope is (7-2)/(3-1) = 5/2. Then substitute this slope and one of the points into y = mx + b to solve for b: 2 = (5/2)(1) + b, so b = -1/2. The final equation is y = (5/2)x - 1/2.

There are two methods for writing a line with a known slope through a point. Using the slope -3/4 and point (1,5), we can use the point-slope form: y - y₁ = m$x - x₁$. This gives us y - 5 = (-3/4)$x - 1$, which simplifies to y = (-3/4)x - 17/4.

💡 The point-slope form $y - y₁ = m(x - x₁)$ is often easier to use when you know a point and slope. You can always convert to slope-intercept form afterward if needed.

Distance, Midpoint, and Line Equations

The distance formula lets you find how far apart two points are on a coordinate plane: d = √$(y₂ - y₁)² + (x₂ - x₁)²$. This is based on the Pythagorean theorem and works for any two points regardless of their position.

If you need to find the middle of a line segment, the midpoint formula gives you the coordinates: M = $(x₁ + x₂)/2, (y₁ + y₂)/2$. This formula simply averages the x-coordinates and y-coordinates of the two endpoints.

When finding a line equation, your approach depends on what information you have. With two points (-5,1) and (3,4), first find the slope: m = (4-1)/(3-(-5)) = 3/8. Then use either point-slope form or solve for b in the slope-intercept form. Both methods should lead to the same equation: y = (3/8)x + (17/8).

💡 When working with lines, you can check your equation by verifying that both of your original points satisfy the equation you've derived. This is a great way to catch calculation errors!

Special Line Relationships and Quadratic Equations

Parallel lines have the same slope but never intersect. Perpendicular lines meet at a 90° angle and have slopes that are negative reciprocals of each other $if one slope is m, the other is -1/m$.

Quadratic equations can be written in the standard form ax² + bx + c = 0. When solving these equations, you have several approaches. If the equation factors easily, like x² - 7x - 30 = 0 into $x-10$$x+3$ = 0, you can set each factor equal to zero to find x = 10 or x = -3.

For equations that don't factor easily, use the completing the square method. Start with x² + 10x - 4 = 0, rearrange to x² + 10x = 4, then add $b/2$² = 25 to both sides: x² + 10x + 25 = 4 + 25. This gives you $x+5$² = 29, so x = -5 ± √29.

💡 When completing the square, remember to take half the coefficient of x, square it, and add that value to both sides of the equation. This transforms your expression into a perfect square trinomial plus a constant.

Solving Quadratics with Multiple Methods

The quadratic formula x = $-b ± √(b² - 4ac)$/(2a) works for any quadratic equation in the form ax² + bx + c = 0. It's your go-to method when other approaches seem difficult.

Let's compare the three main methods for solving quadratics. Factoring is quickest when possible: for x² + 4x + 3 = 0, we factor to $x+1$$x+3$ = 0, giving solutions x = -1 and x = -3.

Completing the square works for any quadratic: for x² + (3/2)x - 4 = 0, rearrange to x² + (3/2)x = 4, add (3/4)² = 9/16 to both sides, and get $x + 3/4$² = 73/16. Taking the square root yields x = -3/4 ± √73/4.

The quadratic formula is most efficient for complex equations: for 11x² - 3x + 2 = 0, identify a = 11, b = -3, c = 2 and substitute into the formula. Here, the discriminant $b² - 4ac$ equals -79, indicating there are no real solutions.

💡 The discriminant $b² - 4ac$ tells you the nature of the solutions: if positive, you have two different real solutions; if zero, one repeated real solution; if negative, two complex solutions.

Functions: Definition and Evaluation

A function is a rule that assigns exactly one output to each input. Think of it like a vending machine—you put in a selection, and you get exactly one item out. If one input could produce multiple different outputs, it's not a function.

In mathematics, we often write functions using notation like f(x) = 2x + 1, where x is the input and f(x) is the output. To evaluate a function, simply substitute the input value for x. For example, with f(x) = x³ - 2x² + 1, we find f(0) = 0³ - 2(0)² + 1 = 1.

You can evaluate a function for any valid input. For f(x) = x³ - 2x² + 1, we calculate f(1) = 1³ - 2(1)² + 1 = 0, f(-1) = (-1)³ - 2(-1)² + 1 = -2, and f(2) = 2³ - 2(2)² + 1 = 1. You can even use variables as inputs, like f(w) = w³ - 2w² + 1.

💡 The vertical line test helps determine if a graph represents a function: if any vertical line intersects the graph more than once, then it's not a function, because one input would be producing multiple outputs.