1.2 Algebraic Operations, Equations, and Inequalities

Fundamentals of Algebraic Expressions

Algebra represents numerical relationships using variables, which are letters that stand in for unknown values. An algebraic expression consists of terms, which are combinations of numbers (coefficients) and variables. For example, in the expression 3x^2 + 5x - 7, there are three terms: 3x^2, 5x, and the constant -7. Simplifying expressions is the first step in solving equations. This is achieved by combining "like terms"—terms that have the identical variable configuration. You can combine 4x and 6x to get 10x, but you cannot combine 4x with 6x^2. The distributive property is another critical tool for simplifying expressions. It states that a(b + c) = ab + ac. Distributing a negative sign across parentheses is a common pitfall; -2(x - 4) simplifies to -2x + 8, not -2x - 8.

Solving Linear Equations

A linear equation is an equation of the first degree, meaning it contains no exponents greater than one on its variables. The fundamental rule of solving equations is that whatever operation you perform on one side of the equal sign, you must perform on the other to maintain balance. To solve for x in the equation 3x + 12 = 27, you first isolate the term with the variable by subtracting 12 from both sides, leaving 3x = 15. Then, divide both sides by 3 to find x = 5. Some equations require variables to be collected on one side first. For 5x - 4 = 2x + 11, subtract 2x from both sides to get 3x - 4 = 11, add 4 to get 3x = 15, and solve to find x = 5.

Systems of Linear Equations

Often, ASTB-E problems involve two variables and two distinct equations, known as a system of equations. There are two primary algebraic methods to solve them: substitution and elimination.

In the substitution method, you solve one equation for one variable and substitute that expression into the other equation. For a system containing x + y = 10 and 2x - y = 8, you can solve the first for x to get x = 10 - y. Substituting this into the second equation yields 2(10 - y) - y = 8, which simplifies to 20 - 3y = 8. Solving this gives -3y = -12, or y = 4. Substituting y = 4 back into x = 10 - y gives x = 6.

The elimination method involves adding or subtracting the equations to cancel out a variable. Taking the same system, simply adding the two equations together (x + y = 10 and 2x - y = 8) eliminates the y variable entirely, leaving 3x = 18, which immediately gives x = 6. This method is often much faster and less error-prone when equations are already aligned in standard form.

Quadratic Equations

Quadratic equations are second-degree polynomials, typically written in the standard form ax^2 + bx + c = 0. Unlike linear equations, quadratics usually have two solutions. The most direct way to solve a quadratic is by factoring. To factor x^2 - 5x + 6 = 0, you must find two numbers that multiply to +6 and add to -5. Those numbers are -2 and -3. Therefore, the factored form is (x - 2)(x - 3) = 0. Setting each factor to zero gives the solutions x = 2 and x = 3.

If a quadratic cannot be easily factored, you must use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a. While more tedious without a calculator, the ASTB-E will typically provide numbers that simplify cleanly. The term b^2 - 4ac under the radical is called the discriminant. If the discriminant is positive, there are two real solutions; if zero, one real solution; if negative, there are no real solutions.

Inequalities

Inequalities operate much like equations, utilizing symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). You solve them using the same algebraic steps as equations, with one vital exception: whenever you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign. For instance, solving -3x > 12 requires dividing by -3. Doing so flips the sign, yielding x < -4. Failing to flip the sign is one of the most common mistakes on the exam. Inequalities can also be graphed on a number line, where an open circle represents strict inequality (< or >) and a closed, filled circle represents inclusive inequality (≤ or ≥).

Real-world Algebraic Applications

Algebra on the ASTB-E is frequently presented in word problems. You might be asked to maximize profit, find the dimensions of a rectangle given its perimeter, or determine relative ages. To translate word problems into algebraic equations, use these common keywords:

• Addition: "sum," "increased by," "more than," "combined," "total"
• Subtraction: "difference," "decreased by," "less than," "reduced by"
• Multiplication: "product," "times," "of" (e.g., "half of the fleet")
• Division: "quotient," "per," "out of," "ratio of"
• Equals: "is," "was," "will be," "results in," "amounts to"

For age problems, create a table representing the present and future or past ages of the individuals. If "John is twice as old as Mary, and in 5 years the sum of their ages will be 34," you can set Mary’s age to x and John’s to 2x. In 5 years, Mary will be x + 5 and John will be 2x + 5. The equation becomes (x + 5) + (2x + 5) = 34, which simplifies to 3x + 10 = 34, meaning 3x = 24 and x = 8. Therefore, Mary is 8 and John is 16.