Precalculus Study Guide: Equations, Inequalities, and Complex Numbers

Ch. 1 - Equations and Inequalities

Solving Linear Equations

Linear equations are fundamental in algebra and precalculus, involving expressions where the variable appears to the first power and is not multiplied by itself. Solving these equations requires systematic application of algebraic operations to isolate the variable.

• Definition: A linear equation is an equation of the form ax + b = c, where x is the unknown variable.

• Key Steps:

  1. Distribute constants across parentheses.

  2. Combine like terms on each side.

  3. Group terms with x and constants on opposite sides.

  4. Isolate x (solve for x).

  5. Check the solution by substituting x 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 clear the denominators by multiplying both sides by the least common denominator (LCD).

• Key Steps:

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

  2. Proceed as with standard linear equations.

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

  • LCD is 12. Multiply both sides by 12:

  • $3(x+2) - 4x = 1$

  • $3x + 6 - 4x = 1$

  • $-x = -5$

  • $x = 5$

Categorizing Linear Equations

Linear equations can be classified based on the number and type of solutions they possess.

• Conditional Equation: Has one solution. Example: $2x + 4 = 10$

• Identity: True for all real numbers. Example: $x + 5 = x + 2 + 3$

• Inconsistent Equation: Has no solution. Example: $x = x + 4$

• Solution Set: The set of values that satisfy the equation.

Solving Rational Equations

Definition and Method

A rational equation is an equation containing one or more rational expressions (fractions with variables in the denominator). Solutions must not make any denominator zero.

• Key Steps:

  1. Determine restrictions by setting denominators equal to zero.

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

  3. Solve the resulting linear equation.

  4. Check solutions against restrictions.

• Example: Solve $\frac{x}{x-1} = \frac{7}{6}$

  • Restriction: $x \neq 1$

  • Cross-multiply: $6x = 7(x-1)$

  • $6x = 7x - 7$

  • $-x = -7$

  • $x = 7$

The Imaginary Unit and Complex Numbers

Square Roots of Negative Numbers

Square roots of negative numbers are not real. The imaginary unit i is defined as $i = \sqrt{-1}$, allowing us to express these roots.

• Example: $\sqrt{-75} = \sqrt{75} \cdot i$

• General Formula: $\sqrt{-b} = i\sqrt{b}$, where $b$ is positive.

Powers of i

Powers of i cycle every four exponents:

• $i^1 = i$

• $i^2 = -1$

• $i^3 = -i$

• $i^4 = 1$

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

• Example: $i^{22} = (i^4)^5 \cdot i^2 = 1^5 \cdot (-1) = -1$

Complex Numbers

A complex number is an expression of the form $a + bi$, 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 distributive property or FOIL, and simplify $i^2$ to -1.

• Example: $(3 + 8i)^2 = (3 + 8i)(3 + 8i) = 9 + 48i + 64i^2 = 9 + 48i - 64 = -55 + 48i$

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 conjugate: $\frac{6 + i}{4 - 2i} \cdot \frac{4 + 2i}{4 + 2i}$

  • Expand and simplify.

Quadratic Equations

Standard Form and Factoring

A quadratic equation is of the form $ax^2 + bx + c = 0$. Solutions are called roots or zeros.

• Factoring: Write in standard form, factor completely, set each factor to zero, and solve for $x$.

• Example: $x^2 + x - 6 = 0$ factors to $(x + 3)(x - 2) = 0$, so $x = -3$ or $x = 2$.

Square Root Property

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

• Example: $(x - 1)^2 = 5$ gives $x - 1 = \pm \sqrt{5}$, so $x = 1 \pm \sqrt{5}$

Completing the Square

Transform $ax^2 + bx + c = 0$ into $(x + d)^2 = k$ by adding $\left(\frac{b}{2}\right)^2$ to both sides.

• Example: $x^2 + 6x = -7$; add $9$to both sides:$x^2 + 6x + 9 = 2$;$(x + 3)^2 = 2$

The Quadratic Formula

The quadratic formula solves any quadratic equation:

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

• Example: $x^2 + 2x - 3 = 0$; $a = 1$, $b = 2$, $c = -3$

• $x = \frac{-2 \pm \sqrt{4 + 12}}{2} = \frac{-2 \pm \sqrt{16}}{2} = \frac{-2 \pm 4}{2}$

• So $x = 1$ or $x = -3$

The Discriminant

The 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

• Example: For $x^2 + 8x + 16 = 0$, $b^2 - 4ac = 64 - 64 = 0$ (one real solution)

Choosing a Method to Solve Quadratic Equations

Depending on the form and coefficients, select the most efficient method: factoring, square root property, completing the square, or quadratic formula.

• Factoring: Use when factors are obvious.

• Square Root Property: Use when $b = 0$.

• Completing the Square: Use when $a = 1$ and $b$ is even.

• Quadratic Formula: Use when factoring is difficult or uncertain.

Linear Inequalities

Interval Notation

Interval notation is a compact way to express solution sets for inequalities.

• Closed Interval: $[a, b]$ means $a \leq x \leq b$

• Open Interval: $(a, b)$ means $a < x < b$

• Half-Open Interval: $[a, b)$ or $(a, b]$

• Infinite Intervals: $(-\infty, a)$ or $(a, \infty)$

Solving Linear Inequalities

Linear inequalities are solved similarly to linear equations, but with an inequality symbol. When multiplying or dividing by a negative number, reverse the inequality sign.

• Example: $2x + 12 > 19$; $2x > 7$; $x > 3.5$

• Interval Notation: $(3.5, \infty)$

Fractions and Variables on Both Sides

Clear fractions by multiplying by the LCD, then solve as usual. Express the solution set in interval notation.

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

• Multiply both sides by 15: $5(x+1) \geq 3(3+2x)$

• $5x + 5 \geq 9 + 6x$

• $5x + 5 - 6x \geq 9$

• $-x + 5 \geq 9$

• $-x \geq 4$

• $x \leq -4$

• Interval Notation: $(-\infty, -4]$