Chapter 1 Study Guide: Equations, Graphs, and Complex Numbers

Equations and Inequalities

Plotting Points in the Rectangular Coordinate System

The rectangular (Cartesian) coordinate system is used to graphically represent points, lines, and curves. Each point is identified by an ordered pair (x, y), where x is the horizontal coordinate and y is the vertical coordinate.

• Quadrants: The plane is divided into four quadrants.

• Plotting: To plot a point, move x units along the horizontal axis and y units along the vertical axis.

• Example: The point (3, -2) is 3 units right and 2 units down from the origin.

Graphing Linear Equations

A linear equation in two variables can be written in the form $Ax + By = C$ or $y = mx + b$ (slope-intercept form).

• Slope (m): Measures the steepness of the line.

• Y-intercept (b): The point where the line crosses the y-axis.

• Graphing Steps:

  1. Find the y-intercept (set x = 0).

  2. Use the slope to find another point.

  3. Draw a straight line through the points.

• Example: For $y = 2x + 1$, the y-intercept is 1, and the slope is 2.

Finding x- and y-Intercepts

The x-intercept is where the graph crosses the x-axis (set y = 0). The y-intercept is where the graph crosses the y-axis (set x = 0).

• Example: For $2x + 3y = 6$:

  • x-intercept: Set y = 0, $2x = 6 \Rightarrow x = 3$

  • y-intercept: Set x = 0, $3y = 6 \Rightarrow y = 2$

Interpreting Real World Data from Graphs

Graphs can be used to model and interpret real-world situations, such as cost, distance, or population over time.

• Key Skills: Identify trends, make predictions, and estimate values from the graph.

• Example: A line graph showing temperature change over days can be used to predict future temperatures.

Solving Linear Equations

A linear equation is an equation of the first degree (highest exponent of the variable is 1).

• General Form: $ax + b = c$

• Steps:

  1. Combine like terms.

  2. Isolate the variable.

  3. Solve for the variable.

• Example: $3x - 5 = 10 \Rightarrow 3x = 15 \Rightarrow x = 5$

Solving Linear Equations with Fractions

To solve equations with fractions, clear denominators by multiplying both sides by the least common denominator (LCD).

• Example: $\frac{1}{2}x + \frac{1}{3} = 1$

• Multiply both sides by 6 (LCD): $3x + 2 = 6 \Rightarrow 3x = 4 \Rightarrow x = \frac{4}{3}$

Applications Involving Linear Models

Linear equations can model real-world relationships, such as cost, distance, or profit.

• Example: If a taxi charges $3 plus $2 per mile, the cost for x miles is$C = 2x + 3$.

Solving a Formula for a Variable

To solve for a specific variable in a formula, isolate that variable using algebraic operations.

• Example: Solve $A = lw$ for $w$: $w = \frac{A}{l}$

Complex Numbers

Addition and Subtraction of Complex Numbers

A complex number is of the form $a + bi$, where $i = \sqrt{-1}$.

• Add/Subtract: Combine real parts and imaginary parts separately.

• Example: $(3 + 2i) + (1 - 5i) = 4 - 3i$

Multiplication and Division of Complex Numbers

• Multiplication: Use distributive property and $i^2 = -1$.

• Example: $(2 + 3i)(1 - 4i) = 2 - 8i + 3i - 12i^2 = 2 - 5i + 12 = 14 - 5i$

• Division: Multiply numerator and denominator by the conjugate of the denominator.

• Example: $\frac{3 + 2i}{1 - i}$, multiply by $1 + i$: $\frac{(3 + 2i)(1 + i)}{(1 - i)(1 + i)} = \frac{3 + 3i + 2i + 2i^2}{1 - i^2} = \frac{3 + 5i - 2}{1 + 1} = \frac{1 + 5i}{2}$

Quadratic Equations

Solving by Factoring

Set the equation to zero, factor, and use the zero product property.

• Example: $x^2 - 5x + 6 = 0 \Rightarrow (x - 2)(x - 3) = 0 \Rightarrow x = 2, 3$

Solving by Completing the Square

Rewrite the equation in the form $(x + p)^2 = q$ and solve for x.

• Steps:

  1. Move constant to the other side.

  2. Add $(\frac{b}{2})^2$ to both sides.

  3. Take square root of both sides.

• Example: $x^2 + 6x + 5 = 0 \Rightarrow x^2 + 6x = -5$ Add $9$to both sides:$x^2 + 6x + 9 = 4(x + 3)^2 = 4 \Rightarrow x + 3 = \pm 2 \Rightarrow x = -1, -5$

Solving by the Quadratic Formula

The quadratic formula solves $ax^2 + bx + c = 0$:

• $x = \frac{ -b \pm \sqrt{b^2 - 4ac} }{2a}$

• Example: $2x^2 + 3x - 2 = 0$ $x = \frac{ -3 \pm \sqrt{3^2 - 4 \cdot 2 \cdot (-2)} }{2 \cdot 2 } = \frac{ -3 \pm \sqrt{9 + 16} }{4 } = \frac{ -3 \pm 5 }{4 }$ $x = \frac{2}{4} = 0.5$ or $x = \frac{-8}{4} = -2$

Using the Discriminant

The discriminant is $b^2 - 4ac$ in the quadratic formula. It determines the number and type of solutions:

• If $b^2 - 4ac > 0$: Two real solutions

• If $b^2 - 4ac = 0$: One real solution

• If $b^2 - 4ac < 0$: Two complex solutions

Solving Quadratic Equations by Any Method

Choose the most efficient method (factoring, completing the square, or quadratic formula) based on the equation.

Radical Equations

Solving Radical Equations

A radical equation contains variables inside a radical (usually a square root). To solve:

• Isolate the radical on one side.

• Raise both sides to the appropriate power to eliminate the radical.

• Solve the resulting equation.

• Check all solutions in the original equation (extraneous solutions may occur).

• Example: $\sqrt{x + 3} = x - 1$ Square both sides: $x + 3 = (x - 1)^2$ $x + 3 = x^2 - 2x + 1$ $0 = x^2 - 3x - 2$ $(x - 4)(x + 1) = 0 \Rightarrow x = 4, -1$ Check both in the original equation.

Graphical Solutions and Intervals

Intersections and Unions of Intervals

Intervals can be represented on a number line. The intersection of intervals is where they overlap; the union is all points in either interval.

• Example: Intersection of [1, 5] and [3, 7] is [3, 5]. Union is [1, 7].

Solving Linear Inequalities

Linear inequalities are solved similarly to equations, but when multiplying or dividing by a negative number, reverse the inequality sign.

• Example: $2x - 3 < 7 \Rightarrow 2x < 10 \Rightarrow x < 5$

• Example: $-3x > 6 \Rightarrow x < -2$ (inequality sign reverses)