Key Precalculus Concepts: Properties of Inequality, Quadratic Equations, and Factoring

Properties of Inequality

Basic Properties and Rules

Inequalities are mathematical statements that compare the relative size of two values. Understanding their properties is essential for solving equations and analyzing functions in precalculus.

• Addition Property: If $a < b$, then $a + c < b + c$ for any real number $c$.

• Multiplication Property (Positive): If $a < b$ and $c > 0$, then $ac < bc$.

• Multiplication Property (Negative): If $a < b$ and $c < 0$, then $ac > bc$ (the inequality reverses).

• Replacement: The same properties apply if $<$ is replaced with $>$.

• Important Note: Always reverse the direction of the inequality when multiplying or dividing both sides by a negative number.

Example: If $-2 < 3$, then $-2 \times (-1) > 3 \times (-1)$, so $2 > -3$.

Quadratic Equations and the Quadratic Formula

Solving Quadratic Equations

Quadratic equations are equations of the form $ax^2 + bx + c = 0$. The quadratic formula provides a systematic way to find their solutions.

• Quadratic Formula: The solutions to $ax^2 + bx + c = 0$ are given by: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Discriminant: The expression under the square root, $b^2 - 4ac$, is called the discriminant. It determines the nature of the solutions:

  • If $b^2 - 4ac > 0$, there are two distinct real solutions.

  • If $b^2 - 4ac = 0$, there is one real solution (a repeated root).

  • If $b^2 - 4ac < 0$, there are two complex solutions.

• Square Roots of Negative Numbers: For $\sqrt{-a}$, where $a > 0$, use $\sqrt{-a} = i\sqrt{a}$, where $i$ is the imaginary unit.

Example: Solve $4x^2 - 3x + 5 = 0$ using the quadratic formula.

• Identify coefficients: $a = 4$, $b = -3$, $c = 5$.

• Compute discriminant: $(-3)^2 - 4 \times 4 \times 5 = 9 - 80 = -71$.

• Apply formula: $x = \frac{3 \pm \sqrt{-71}}{8}$ Since $\sqrt{-71} = i\sqrt{71}$, $x = \frac{3}{8} \pm \frac{i\sqrt{71}}{8}$

Additional info: The example demonstrates how to handle complex solutions when the discriminant is negative.

Factoring: Difference of Squares

Factoring Patterns

Factoring is a method of rewriting expressions as products of simpler expressions. The difference of squares is a common pattern in precalculus.

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

• Application: This formula is used to factor expressions where two squared terms are subtracted.

Example: Factor $x^2 - 9$. $x^2 - 9 = (x + 3)(x - 3)$

Summary Table: Properties of Inequality

Property	Rule	Example
Addition	If $a < b$, then $a + c < b + c$	If $2 < 5$, then $2 + 3 < 5 + 3$ ($5 < 8$)
Multiplication (Positive)	If $a < b$ and $c > 0$, then $ac < bc$	If $1 < 4$ and $c = 2$, then $2 < 8$
Multiplication (Negative)	If $a < b$ and $c < 0$, then $ac > bc$	If $1 < 4$ and $c = -2$, then $-2 > -8$
Replacement	Same rules apply for $>$	If $a > b$, then $a + c > b + c$