Algebra Practice Problems: Key Precalculus Topics

Rational Equations

Solving Rational Equations

Rational equations involve fractions whose numerators and/or denominators contain polynomials. Solving these equations typically requires finding a common denominator and ensuring that solutions do not make any denominator zero.

• Key Point 1: Multiply both sides by the least common denominator (LCD) to clear fractions.

• Key Point 2: Check for extraneous solutions by substituting back into the original equation.

• Example: Solve $\frac{4}{x} + \frac{3}{x-1} = \frac{7(x)}{x(x-1)}$

Absolute Value Equations & Inequalities

Solving Absolute Value Equations

Absolute value equations require considering both the positive and negative cases of the expression inside the absolute value.

• Key Point 1: For $|ax + b| = c$, write two equations: $ax + b = c$ and $ax + b = -c$.

• Key Point 2: For inequalities, split into two cases and solve each.

• Example: Solve $|2x - 5| = 12$

Linear & Polynomial Equations

Solving Linear and Polynomial Equations

Linear equations have the form $ax + b = 0$. Polynomial equations may be quadratic or higher degree and can often be solved by factoring or using the quadratic formula.

• Key Point 1: Set the equation to zero and factor if possible.

• Key Point 2: For quadratics, use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Example: Solve $2x^2 + 3x - 2 = 0$

Radical Equations

Solving Radical Equations

Radical equations contain variables inside a root. Isolate the radical and then raise both sides to the appropriate power to eliminate the root.

• Key Point 1: Isolate the radical on one side of the equation.

• Key Point 2: Raise both sides to the power that matches the root (e.g., square both sides for square roots).

• Key Point 3: Check for extraneous solutions.

• Example: Solve $\sqrt{2x + 3} = x - 1$

Inequalities

Solving Linear and Quadratic Inequalities

Inequalities involve finding the set of values that satisfy a given relationship. For quadratics, test intervals between critical points.

• Key Point 1: For linear inequalities, isolate the variable as in equations.

• Key Point 2: For quadratic inequalities, find zeros and test intervals.

• Example: Solve $2x - 5 < 7$

Quadratic Equations in Special Forms

Factoring and Solving Quadratics

Some quadratics can be solved by factoring, especially when they are in the form $ax^2 + bx + c = 0$.

• Key Point 1: Factor the quadratic if possible.

• Key Point 2: Set each factor equal to zero and solve for $x$.

• Example: Solve $x^2 - 9x + 18 = 0$

Complex Numbers

Introduction to Complex Numbers

Complex numbers are numbers of the form $a + bi$, where $i = \sqrt{-1}$.

• Key Point 1: $i^2 = -1$

• Key Point 2: Operations with complex numbers include addition, subtraction, multiplication, and division.

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

Word / Finance Problems

Applications in Finance

Word problems in algebra often involve translating real-world scenarios into equations. Finance problems may involve investments, interest, or total value calculations.

• Key Point 1: Define variables clearly.

• Key Point 2: Set up equations based on the problem context.

• Example: If $50,000 is invested in stocks and bonds, write an equation for the total investment.