GRE Prep: Algebra (Linear & Quadratic Equations, Inequalities, Functions)

GRE – Algebra (Linear & Quadratic Equations, Inequalities, Functions)

GRE Study Guide – Algebra (Linear & Quadratic Equations, Inequalities, Functions)

What This Is

Algebra on the GRE covers solving linear and quadratic equations, handling systems of equations, and working with inequalities and function notation. These skills are the backbone of the Quantitative Reasoning section because every “word‑problem” can be reduced to an algebraic expression.
Typical test question: “If (2x+5=13), what is the value of (x^2)?” – you must solve the linear equation for (x) and then square the result.

Key Terms & Rules

• Linear Equation: An equation of the form (ax+b=0) or (y=mx+b). Example: (3x-7=2).
• Quadratic Equation: An equation that can be written as (ax^2+bx+c=0). Use the quadratic formula (x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}).
• Factoring: Rewriting a quadratic as ((px+q)(rx+s)=0). Helpful when the discriminant is a perfect square.
• Completing the Square: Transform (ax^2+bx) into (a\bigl(x+\frac{b}{2a}\bigr)^2-\frac{b^{2}}{4a}). Used when the quadratic formula feels cumbersome.
• System of Linear Equations: Two or more linear equations solved simultaneously (substitution, elimination, or matrix method).
• Inequality Direction Flip: Multiply or divide both sides by a negative number → reverse the inequality sign ( (>) ↔ ( <) ).
• Absolute‑Value Equation/Inequality: (|x|=k) gives (x=k) or (x=-k); (|x|<k) yields (-k<x<k).
• Function Notation: (f(x)=) expression; evaluate by substituting the given (x). Inverse functions are denoted (f^{-1}(x)).
• Domain & Range Restrictions: For rational functions, exclude values that make the denominator zero; for square‑root functions, require the radicand (\ge0).
• Quadratic Inequality: Solve the corresponding equation, then test a point in each interval; the solution set is a union of intervals.
• Slope‑Intercept Form: (y=mx+b); the slope (m) tells you rise over run—critical for quick comparison questions.
• Vertex Form of a Parabola: (y=a(x-h)^2+k); the vertex ((h,k)) gives the minimum/maximum value instantly.

Step‑by‑Step / Process Flow

1. Read the stem carefully – note whether the problem asks for a value, the set of possible values, or a comparison.
2. Translate words into an equation – assign a variable, write the algebraic relationship, and keep units consistent.
3. Choose the simplest solving method – factor if the quadratic factors nicely; otherwise apply the quadratic formula or complete the square.
4. Check for extraneous solutions – plug each candidate back into the original equation/inequality (especially with absolute values or rational expressions).
5. Select the answer – eliminate choices that violate domain restrictions, sign flips, or the required inequality direction.

Common Mistakes

• Mistake: Forgetting to flip the inequality sign when dividing by a negative number.
  Correction: Explicitly write “÷ (–3) → reverse (>) to (<)” and double‑check the final inequality.

• Mistake: Assuming a quadratic always has two distinct real roots.
  Correction: Compute the discriminant (b^{2}-4ac); if it’s zero, there’s one repeated root; if negative, no real solutions (answer may be “none of the above”).

• Mistake: Ignoring domain restrictions for rational or square‑root functions, leading to illegal values.
  Correction: Write the denominator ≠ 0 and radicand ≥ 0 before solving; discard any solution that violates these conditions.

• Mistake: Treating (|x|=k) as a single solution (x=k).
  Correction: Remember absolute value yields two possibilities: (x=k) or (x=-k).

• Mistake: Substituting the wrong variable into a function (e.g., using (y) instead of the given (x)).
  Correction: Keep a clear “variable map” on the side: “Given (x=4) → compute (f(4)).”

Exam Insights

1. Most‑tested concept: Solving linear equations with one unknown; the GRE often disguises them inside word problems.
2. Quadratic trap: The answer choices may include the discriminant value instead of the actual roots—always finish the calculation.
3. Inequality nuance: “Greater than or equal to” vs. “strictly greater than” changes the endpoint inclusion; watch for brackets in answer sets.
4. Function shortcut: For a linear function (f(x)=mx+b), the difference between two outputs equals (m) times the difference between inputs—useful for Quantitative Comparison items.

Quick Check Questions

1. Multiple‑Choice (Algebraic Manipulation)
   
   If (3x-4=2), what is (x^2)?
   A) 4 B) 9 C) 16 D) 25 E) None of the above
   Answer: B. Solve: (3x=6) → (x=2); (x^2=4). (Oops! Actually (x=2) → (x^2=4). The correct answer is A. Explanation: Quick arithmetic; many test‑takers mis‑read the sign.

2. Quantitative Comparison
   
   Statement I: The larger root of (x^2-5x+6=0) is 3.
   
   Statement II: The product of the roots of the same equation is 6.
   A) Statement I alone is sufficient.
   B) Statement II alone is sufficient.
   C) Both together are sufficient.
   D) Each alone is sufficient, but together is not needed.
   E) Neither alone nor together is sufficient.
   Answer: A. The quadratic factors to ((x-2)(x-3)=0); the larger root is clearly 3, so Statement I alone gives the answer.

3. Data‑Sufficiency (Inequality)
   
   If ( |2y-7| < 5), is (y) between 1 and 6?
   A) Yes B) No C) Cannot be determined D) Statement 1 alone is sufficient E) Statement 2 alone is sufficient
   Answer: A. Solving (-5 < 2y-7 < 5) → (2 < 2y < 12) → (1 < y < 6). The inequality holds, so the statement is true.

Last‑Minute Cram Sheet (10 One‑Liners)

1. Linear equation: (ax+b=0) → (x=-\frac{b}{a}).
2. Quadratic formula: (x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}).
3. Discriminant rule: (b^{2}-4ac>0) → 2 real roots; = 0 → 1 repeated root; < 0 → no real roots.
4. Inequality flip: Multiplying/dividing by a negative always reverses (>) ↔ (<). ⚠️
5. Absolute value: (|x|=k) → (x=k) or (x=-k); (|x|<k) → (-k<x<k).
6. Domain check: Denominator ≠ 0; radicand ≥ 0. Discard any solution violating these. ⚠️
7. Vertex of parabola: (y=a(x-h)^2+k) → vertex ((h,k)); gives min (if (a>0)) or max (if (a<0)).
8. System of two linear equations: Add/subtract to eliminate one variable → solve for the other.
9. Function evaluation: (f(x)=mx+b) → (f(c)=mc+b). Remember to substitute the given value, not the variable name.
10. Quick QC tip: If a problem asks for “the greatest possible integer value of (x)”, solve the inequality, then take the floor of the upper bound. ⚠️

Good luck—master these algebraic tools and you’ll breeze through the GRE Quantitative Reasoning section!