College Algebra: Equations, Inequalities, and Complex Numbers

Ch. 1 - Equations and Inequalities

Solving Linear Equations

Linear equations are algebraic equations in which each term is either a constant or the product of a constant and a single variable. The general form is ax + b = c. Solving these equations involves isolating the variable using inverse operations.

• Definition: A linear equation is an equation of the form ax + b = c, where a, b, and c are constants.

• Steps to Solve:

  1. Distribute constants (if necessary).

  2. Combine like terms on each side.

  3. Group terms with x on one side and constants on the other.

  4. Isolate x by performing inverse operations.

  5. Check the solution by substituting back into the original equation.

• Example: Solve $2(x-3) = 0$

  • Distribute: $2x - 6 = 0$

  • Add 6: $2x = 6$

  • Divide by 2: $x = 3$

Linear Equations with Fractions

When linear equations contain fractions, it is often helpful to eliminate denominators by multiplying both sides by the least common denominator (LCD).

• Steps:

  1. Multiply both sides by the LCD to clear fractions.

  2. Proceed as with standard linear equations.

• Example: Solve $\frac{1}{4}(x+2) - \frac{1}{3}x = \frac{1}{12}$

  • Multiply both sides by 12 (LCD): $3(x+2) - 4x = 1$

  • Expand: $3x + 6 - 4x = 1$

  • Combine like terms: $-x + 6 = 1$

  • Isolate $x$: $-x = -5 \implies x = 5$

Categorizing Linear Equations

Linear equations can be classified based on the number of solutions:

• Conditional Equation: Has exactly one solution (e.g., $2x + 4 = 10$).

• Identity: True for all real numbers (e.g., $x + 5 = x + 5$).

• Inconsistent Equation: Has no solution (e.g., $x = x + 4$).

Complex Numbers

Introduction to Complex Numbers

Complex numbers extend the real numbers by including the imaginary unit $i$, where $i = \sqrt{-1}$. The standard form is $a + bi$, where $a$ is the real part and $b$ is the imaginary part.

• Standard Form: $a + bi$

• Example: $4 - 3i$ has real part $4$and imaginary part$-3$.

Operations with Complex Numbers

Adding and Subtracting

Combine like terms (real with real, imaginary with imaginary).

• Example: $(2 + 8i) - (4 - i) = (2 - 4) + (8i + i) = -2 + 9i$

Multiplying

Use the distributive property (FOIL) and simplify using $i^2 = -1$.

• Example: $(3 + 8i)^2 = (3 + 8i)(3 + 8i) = 9 + 24i + 64i^2 = 9 + 24i - 64 = -55 + 24i$

Complex Conjugates

The conjugate of $a + bi$ is $a - bi$. Multiplying a complex number by its conjugate yields a real number.

• Example: $(2 + 3i)(2 - 3i) = 4 - 9i^2 = 4 + 9 = 13$

Dividing Complex Numbers

To divide by a complex number, multiply numerator and denominator by the conjugate of the denominator to make the denominator real.

• Example: $\frac{6 + i}{4 - 2i}$ Multiply by $\frac{4 + 2i}{4 + 2i}$ and simplify.

Powers of $i$

Powers of $i$ repeat in a cycle of four:

• $i^1 = i$

• $i^2 = -1$

• $i^3 = -i$

• $i^4 = 1$

For higher powers, divide the exponent by 4 and use the remainder to determine the value.

Quadratic Equations

Factoring Quadratic Equations

A quadratic equation is an equation of the form $ax^2 + bx + c = 0$. Factoring is one method to solve for $x$.

• Steps:

  1. Write in standard form: $ax^2 + bx + c = 0$

  2. Factor completely.

  3. Set each factor equal to zero and solve for $x$.

  4. Check solutions in the original equation.

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

The Square Root Property

Used when the quadratic can be written as $(x + a)^2 = k$. Take the square root of both sides to solve for $x$.

• Example: $(x - 1/2)^2 - 5 = 0 \implies (x - 1/2)^2 = 5 \implies x - 1/2 = \pm \sqrt{5} \implies x = 1/2 \pm \sqrt{5}$

Completing the Square

Completing the square rewrites a quadratic in the form $(x + d)^2 = k$ to solve for $x$.

• Steps:

  1. Move the constant to the other side: $x^2 + bx = c$

  2. Add $(b/2)^2$ to both sides.

  3. Factor the left as a perfect square trinomial.

  4. Solve using the square root property.

The Quadratic Formula

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

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

• Discriminant: $b^2 - 4ac$ 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

Equations of Two Variables

Solving and Graphing Two-Variable Equations

Equations with two variables, typically $x$ and $y$, represent relationships between variables and can be graphed as lines or curves in the coordinate plane.

• To check if a point $(x, y)$ satisfies an equation: Substitute $x$ and $y$ into the equation and see if the statement is true.

• Graphing by Plotting Points:

  1. Isolate $y$ (if possible).

  2. Choose several $x$-values and compute corresponding $y$-values.

  3. Plot the points $(x, y)$ and connect them.

Intercepts

The x-intercept is where the graph crosses the x-axis ($y = 0$), and the y-intercept is where it crosses the y-axis ($x = 0$).

• To find intercepts:

  • Set $y = 0$ and solve for $x$ (x-intercept).

  • Set $x = 0$ and solve for $y$ (y-intercept).

Solving Rational Equations

Rational Equations

A rational equation contains a variable in the denominator. To solve, clear denominators and solve the resulting equation, but check for extraneous solutions that make any denominator zero.

• Steps:

  1. Find restrictions by setting denominators equal to zero.

  2. Multiply both sides by the LCD to clear fractions.

  3. Solve the resulting equation.

  4. Check solutions against restrictions.

Linear Inequalities

Interval Notation

Interval notation is a concise way to describe sets of solutions for inequalities.

• Closed Interval: $[a, b]$ includes endpoints $a$ and $b$.

• Open Interval: $(a, b)$ excludes endpoints.

• Half-Open Interval: $[a, b)$ or $(a, b]$ includes one endpoint.

• Infinity: Always use parentheses with $\infty$ or $-\infty$.

Solving Linear Inequalities

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

• Example: Solve $2x + 12 > 19$

  • Subtract 12: $2x > 7$

  • Divide by 2: $x > 3.5$

  • Interval notation: $(3.5, \infty)$

Inequalities with Fractions and Variables on Both Sides

Clear fractions by multiplying both sides by the LCD, then solve as usual. Remember to reverse the inequality if multiplying/dividing by a negative.

• Example: $\frac{1}{3}(x + 1) \geq \frac{1}{5}(3 + 2x)$