Glossary

Absolute Value Equations

Criticality: 3

Equations that contain an expression within absolute value symbols, representing the distance of a number from zero.

Example:

The equation $|3x - 6| = 9$ is an absolute value equation that requires considering two possibilities.

Eliminating the Radical

Criticality: 2

The step where both sides of a radical equation are raised to a power equal to the root's index to remove the radical symbol.

Example:

After isolating $\sqrt{x} = 5$ , you perform eliminating the radical by squaring both sides to find $x=25$ .

Extraneous Solutions

Criticality: 3

Solutions that arise during the solving process but do not satisfy the original equation when substituted back in. They often appear when squaring both sides of an equation.

Example:

If you solve a radical equation and get $x=4$ and $x=1$ , but only $x=4$ works in the original, then $x=1$ is an extraneous solution.

Inverse Operations

Criticality: 2

Operations that undo each other, used to isolate variables or terms in an equation. Examples include addition/subtraction and multiplication/division.

Example:

To get 'x' by itself in $x-7=10$ , you'd use addition, the inverse operation of subtraction.

Isolating the Radical (or Absolute Value)

Criticality: 3

The process of manipulating an equation using inverse operations to get the radical or absolute value term by itself on one side of the equation.

Example:

Before solving $\sqrt{x+2} + 1 = x$ , your first step is isolating the radical to get $\sqrt{x+2} = x-1$ .

Least Common Denominator (LCD)

Criticality: 2

The smallest common multiple of the denominators of a set of fractions, used to clear fractions in rational equations by multiplying every term by it.

Example:

To combine $\frac{1}{x}$ and $\frac{1}{x+2}$ , the Least Common Denominator would be $x(x+2)$ .

Radical Equations

Criticality: 3

Equations that contain a variable under a radical symbol, such as a square root or cube root.

Example:

Solving $\sqrt{2x+5} = 3$ is a classic example of a radical equation you'll encounter on the SAT.

Rational Equations

Criticality: 3

Equations that involve one or more rational expressions, which are fractions with variables in their denominators.

Example:

An equation like $\frac{2}{x+1} + \frac{1}{x} = 1$ is a rational equation because of the variables in the denominators.

Two Cases (for Absolute Value)

Criticality: 3

The method for solving absolute value equations by splitting them into two separate linear equations: one where the expression inside the absolute value equals the positive value, and one where it equals the negative value.

Example:

When solving $|x-5|=3$ , you must consider two cases: $x-5=3$ and $x-5=-3$ .

Verifying Solutions

Criticality: 3

The crucial step of substituting potential solutions back into the *original* equation to confirm their validity and identify any extraneous solutions.

Example:

Always remember to perform verifying solutions for radical and rational equations; it's how you catch tricky extraneous answers.