Algebra

Strategy/Review

Simplifying Expressions

The core skill: combine like terms systematically. Work left to right, grouping terms with the same variable and exponent.

Example: Simplify 3x² + 5x − 2x² + 7x − 4 → Group x² terms: 3x² − 2x² = x² → Group x terms: 5x + 7x = 12x → Constants: −4 → Result: x² + 12x − 4

Fast rearrangement technique: When an expression is messy, circle each "chunk" (each term with its sign) and rearrange mentally before writing.

Solving Linear Equations

The golden rule: whatever you do to one side, do to the other. Your goal is to isolate the variable.

Standard approach:

1. Clear parentheses (distribute).
2. Combine like terms on each side.
3. Move variable terms to one side, constants to the other.
4. Divide by the coefficient.

Example: 3(x − 4) + 2x = 5x − 8 → 3x − 12 + 2x = 5x − 8 → 5x − 12 = 5x − 8 → −12 = −8 ❌ No solution (contradiction). The DAT sometimes includes equations with no solution or infinite solutions — recognize these quickly.

Systems of Equations

Two methods, choose based on the equation structure:

Substitution: Best when one variable is already isolated or easy to isolate. → Solve one equation for x (or y). → Substitute that expression into the other equation. → Solve for the remaining variable. → Back-substitute to find the other.

Elimination: Best when coefficients line up nicely. → Multiply one or both equations so the coefficients of one variable are opposites. → Add the equations to eliminate that variable. → Solve for the remaining variable. → Back-substitute.

Example — Elimination: 2x + 3y = 12 4x − 3y = 6 → Add: 6x = 18 → x = 3 → Substitute: 2(3) + 3y = 12 → 6 + 3y = 12 → 3y = 6 → y = 2 → Solution: (3, 2)

Inequalities

Same rules as equations with ONE critical exception: multiplying or dividing by a NEGATIVE number FLIPS the inequality sign.

Example: −2x + 5 > 11 → −2x > 6 → x < −3 (divided by −2, flipped the sign)

Compound inequalities: −4 ≤ 2x + 1 < 7 → Subtract 1: −5 ≤ 2x < 6 → Divide by 2: −2.5 ≤ x < 3

Absolute Value Equations

The rule: |expression| = k means expression = k OR expression = −k. Always check BOTH cases.

Example: |2x − 3| = 7 → Case 1: 2x − 3 = 7 → 2x = 10 → x = 5 → Case 2: 2x − 3 = −7 → 2x = −4 → x = −2 → Solutions: x = 5 or x = −2

Absolute value inequalities (split into two): |x| < k → −k < x < k |x| > k → x < −k OR x > k

Exponent Rules (Must Be Automatic)

1. xᵃ · xᵇ = xᵃ⁺ᵇ (multiplying same base → add exponents)
2. xᵃ / xᵇ = xᵃ⁻ᵇ (dividing same base → subtract exponents)
3. (xᵃ)ᵇ = xᵃᵇ (power to a power → multiply exponents)
4. x⁰ = 1 (any nonzero base to the 0 power = 1)
5. x⁻ᵃ = 1/xᵃ (negative exponent → reciprocal)
6. x^(1/n) = ⁿ√x (fractional exponent → root)
7. (xy)ᵃ = xᵃyᵃ (power of a product)

Radicals

Key simplification: √(x²) = |x| (the principal root is always nonnegative).

Operations:

• √a · √b = √(ab)
• √a / √b = √(a/b)
• To add/subtract radicals, they must have the same radicand: 2√3 + 5√3 = 7√3, but 2√3 + 5√2 cannot be combined.

Rationalizing denominators: Multiply numerator and denominator by the conjugate or the radical that clears the denominator.

Quadratics

Three solution methods, ordered by preference:

1. Factoring (fastest): If the quadratic factors nicely. x² + 5x + 6 = 0 → (x + 2)(x + 3) = 0 → x = −2 or x = −3

2. Quadratic formula (always works): x = [−b ± √(b² − 4ac)] / 2a Use the discriminant (b² − 4ac) to predict:

• If > 0: two real solutions
• If = 0: one real solution (double root)
• If < 0: no real solutions (complex)

3. Completing the square: Useful when the quadratic is almost a perfect square, or when the answer choices are in vertex form.

Common Traps

Trap 1 — Inequality Sign Forgetfulness. The single most common QR algebra error: dividing by a negative without flipping the sign. Every time you divide or multiply by a negative in an inequality, pause and verify the sign direction.

Trap 2 — Distributing Minus Signs Incorrectly. −(3x − 4) = −3x + 4, not −3x − 4. The minus sign distributes to EVERY term inside the parentheses.

Trap 3 — Extraneous Solutions from Squaring. When you square both sides of an equation (common with radical equations), you may introduce extraneous solutions. Always plug answers back into the original equation.

Trap 4 — Forgetting ± in Square Roots. x² = 16 means x = ±4. The equation has two solutions, even though √16 = 4 (just the principal root). Don't confuse solving x² = k (two solutions) with evaluating √k (one, nonnegative).

Trap 5 — Canceling Terms Instead of Factors. You can cancel FACTORS (things being multiplied), but NOT TERMS (things being added/subtracted). (x² + 3x)/x = x + 3 (cancel factor x). But (x + 3)/x does NOT simplify to 3 — you cannot cancel the x terms.