Equations and Inequalities: Precalculus Study Notes

Equations and Inequalities

Solving Inequalities

Inequalities are mathematical statements that compare two values or expressions using symbols such as > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to). Solving inequalities involves finding the set of values that satisfy the inequality.

• > (greater than): Indicates that one value is larger than another.

• < (less than): Indicates that one value is smaller than another.

• ≥ (greater than or equal to): Indicates that one value is larger than or equal to another.

• ≤ (less than or equal to): Indicates that one value is smaller than or equal to another.

Interval Notation

Interval notation is a concise way to describe sets of real numbers that satisfy inequalities. It uses parentheses and brackets to indicate whether endpoints are included or excluded.

• (a, b): All real numbers x for which a < x < b (endpoints not included).

• [a, b]: All real numbers x for which a ≤ x ≤ b (endpoints included).

• (a, b]: All real numbers x for which a < x ≤ b.

• [a, b): All real numbers x for which a ≤ x < b.

Solving Linear Inequalities

Linear inequalities are similar to linear equations but use inequality signs. The solution is often expressed in interval notation.

• Example: Solve $3x + 2 > 7$ Step 1: Subtract 2 from both sides: $3x > 5$ Step 2: Divide both sides by 3: $x > \frac{5}{3}$ Solution: $(\frac{5}{3}, \infty)$

• Example: Solve $2x - 5 < 3$ Step 1: Add 5 to both sides: $2x < 8$ Step 2: Divide both sides by 2: $x < 4$ Solution: $(-\infty, 4)$

Solving Quadratic Inequalities

Quadratic inequalities involve expressions of the form $ax^2 + bx + c$. Solutions are often found by factoring or using the quadratic formula, then testing intervals.

• Example: Solve $x^2 - 5x + 6 > 0$ Step 1: Factor: $(x - 2)(x - 3) > 0$ Step 2: Find zeros: $x = 2$, $x = 3$ Step 3: Test intervals:

  • $x < 2$: Both factors negative, product positive

  • $2 < x < 3$: One factor positive, one negative, product negative

  • $x > 3$: Both factors positive, product positive

  Solution: $(-\infty, 2) \cup (3, \infty)$

Solving Equations: Linear, Quadratic, Rational, and Radical

Equations are mathematical statements asserting equality. Different types require different methods:

• Linear Equation: $ax + b = 0$ Example: $3x + 2 = 4$ Solution: $x = \frac{2}{3}$

• Quadratic Equation: $ax^2 + bx + c = 0$ Example: $x^2 - 5x + 6 = 0$ Solution: $x = 2, 3$

• Rational Equation: Contains fractions with variables in the denominator. Example: $\frac{2x}{x-1} = 4$ Solution: $x = \frac{2}{3}$

• Radical Equation: Contains square roots or other roots. Example: $\sqrt{x+2} = 3$ Solution: $x = 7$

Factoring Techniques

Factoring is a method used to solve equations and simplify expressions. Common techniques include:

• Difference of Squares: $a^2 - b^2 = (a + b)(a - b)$

• Sum and Difference of Cubes:

  • $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$

  • $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

• Grouping: Used for polynomials with four or more terms.

Grouping Example

• Example: $x^2 + 5x - 6 = 0$ Step 1: Factor: $(x + 6)(x - 1) = 0$ Step 2: Set each factor to zero: $x + 6 = 0$ or $x - 1 = 0$ Solution: $x = -6$ or $x = 1$

Summary Table: Types of Equations and Methods

Type of Equation	General Form	Solution Method
Linear	$ax + b = 0$	Isolate variable
Quadratic	$ax^2 + bx + c = 0$	Factoring, quadratic formula
Rational	$\frac{a}{b} = c$	Clear denominators, solve
Radical	$\sqrt{a} = b$	Isolate radical, square both sides

Additional info:

• Interval notation is essential for expressing solutions to inequalities.

• Factoring is a foundational skill for solving quadratic and higher-order equations.

• Graphical representations, such as the parabola in image_1, help visualize solutions to quadratic inequalities.