Hour 2: Algebra & Equations | 24-Hour GMAT Crash Course

What You'll Learn This Hour

1 Solve linear equations, simultaneous systems, and recognize when a system is inconsistent or has infinite solutions.
2 Factor and apply the quadratic formula to solve any second-degree equation quickly.
3 Handle inequalities and absolute value equations without introducing extraneous solutions.
4 Apply all key exponent rules and simplify complex algebraic expressions under time pressure.

Core Concepts

Linear Equations & Systems

Isolate the variable by performing identical operations on both sides. For a 2×2 system, use substitution or elimination. A system has one solution (lines intersect), no solution (parallel lines), or infinite solutions (same line).

Quadratic Equations

Factor when possible: (x + a)(x + b) = 0 means x = -a or x = -b. Otherwise use the quadratic formula: x = (-b ± √(b²-4ac)) / 2a. The discriminant b²-4ac tells you: positive = 2 real roots, zero = 1 real root, negative = no real roots.

Inequalities

All arithmetic rules apply EXCEPT: multiplying or dividing both sides by a negative number reverses the inequality sign. When the sign of a variable is unknown, split into cases. Never cross-multiply in an inequality without knowing the sign.

Absolute Value

|x| = k means x = k or x = -k (valid only when k ≥ 0). |x| = -1 has no solution. For |x - a| < b, solve -b < x - a < b. For |x - a| > b, solve x - a > b OR x - a < -b. Always check solutions.

Exponent Rules

a^m · a^n = a^(m+n)

Multiply same base → add exponents

a^m / a^n = a^(m-n)

Divide same base → subtract exponents

(a^m)^n = a^(mn)

Power to a power → multiply exponents

a^0 = 1

Any nonzero base to the 0 equals 1

a^(-n) = 1/a^n

Negative exponent → reciprocal

a^(1/n) = ⁿ√a

Fractional exponent → nth root

Visualizing a Linear Equation

The line y = 2x − 3 crosses the y-axis at (0, −3) and the x-axis at (1.5, 0). The slope triangle shows rise = 2, run = 1, confirming slope = 2.

Worked Examples

Example 1 System of Equations

Solve: 3x + 2y = 12 and x − y = 1

Step 1 — Express x in terms of y from equation 2:
x − y = 1 → x = y + 1

Step 2 — Substitute into equation 1:
3(y + 1) + 2y = 12 → 3y + 3 + 2y = 12 → 5y = 9 → y = 9/5

Step 3 — Back-substitute:
x = (9/5) + 1 = 14/5

Answer: x = 14/5, y = 9/5. Verify: 3(14/5) + 2(9/5) = 42/5 + 18/5 = 60/5 = 12. ✓

Example 2 Quadratic Equation

Find all values of x: 2x² − 5x − 3 = 0

Step 1 — Identify a, b, c:
a = 2, b = −5, c = −3

Step 2 — Compute the discriminant:
b² − 4ac = (−5)² − 4(2)(−3) = 25 + 24 = 49

Step 3 — Apply quadratic formula:
x = (5 ± √49) / 4 = (5 ± 7) / 4

Step 4 — Both roots:
x = (5 + 7)/4 = 3 | x = (5 − 7)/4 = −1/2

Answer: x = 3 or x = −1/2

Example 3 Absolute Value Inequality

Solve: |2x − 4| ≤ 6

Step 1 — Set up compound inequality:
−6 ≤ 2x − 4 ≤ 6

Step 2 — Add 4 to all parts:
−6 + 4 ≤ 2x ≤ 6 + 4 → −2 ≤ 2x ≤ 10

Step 3 — Divide by 2 (positive, no flip):
−1 ≤ x ≤ 5

Answer: −1 ≤ x ≤ 5 (closed interval [−1, 5])

GMAT Traps to Avoid

Dividing or multiplying by a negative flips the inequality

If −2x > 6, dividing by −2 gives x < −3, NOT x > −3. This is the #1 algebraic trap on the GMAT.

Squaring both sides can introduce extraneous solutions

If √x = x − 2, squaring gives x = x² − 4x + 4. Both solutions of the quadratic must be checked back in the original equation — one may be extraneous.

|x| = −1 has NO solution

Absolute value is always non-negative. An equation like |3x + 2| = −5 has zero solutions — but the GMAT may list "0" or "no solution" as a trap answer.

Don't confuse "no solution" with "infinite solutions" for systems

2x + y = 4 and 4x + 2y = 9 → parallel lines (no solution). But 2x + y = 4 and 4x + 2y = 8 → same line (infinite solutions). Check the ratio of constants carefully.

Practice Questions

12 GMAT-style questions. Try each before revealing the answer.

Q1. If 3x − 7 = 2x + 5, what is x?

(A) 2(B) 5(C) 10(D) 12(E) 14

Show Answer

Answer: (D) 12
3x − 2x = 5 + 7 → x = 12. Straightforward linear equation — collect like terms.

Q2. Solve x² − 9x + 20 = 0. Which of the following are the solutions?

(A) 2, 10(B) 4, 5(C) −4, −5(D) 3, 7(E) 1, 20

Show Answer

Answer: (B) 4, 5
Factor: (x − 4)(x − 5) = 0. So x = 4 or x = 5. Check: 4 + 5 = 9 ✓, 4 × 5 = 20 ✓.

Q3. If −3x > 12, which of the following must be true?

(A) x > −4(B) x > 4(C) x < −4(D) x < 4(E) x = −4

Show Answer

Answer: (C) x < −4
Dividing by −3 flips the sign: x < 12/(−3) = −4. The trap is choosing (A) by forgetting to flip.

Q4. Solve |x + 3| = 7.

(A) 4 only(B) −10 only(C) 4 and −10(D) 10 and −4(E) No solution

Show Answer

Answer: (C) 4 and −10
Two cases: x + 3 = 7 → x = 4; or x + 3 = −7 → x = −10. Both satisfy the original equation.

Q5. Simplify: (2³ × 2⁵) / 2⁴

(A) 2²(B) 2⁴(C) 2⁶(D) 2¹²(E) 2⁻²

Show Answer

Answer: (B) 2⁴
Numerator: 2³ × 2⁵ = 2⁸. Then 2⁸ / 2⁴ = 2^(8−4) = 2⁴ = 16.

Q6. If 2x + y = 10 and x + 2y = 8, what is x + y?

(A) 4(B) 5(C) 6(D) 7(E) 8

Show Answer

Answer: (C) 6
Add the two equations: 3x + 3y = 18 → x + y = 6. No need to find x and y separately — a key GMAT time-saver.

Q7. How many solutions does |2x − 6| = −4 have?

(A) 0(B) 1(C) 2(D) 3(E) Infinite

Show Answer

Answer: (A) 0
The absolute value of any real expression is always ≥ 0. It can never equal −4. Zero solutions.

Q8. If x² = 25, what are all possible values of x?

(A) 5 only(B) −5 only(C) 5 and −5(D) 25 and −25(E) √5 only

Show Answer

Answer: (C) 5 and −5
x² = 25 → x = ±√25 = ±5. Note: √25 = 5 (principal root), but the equation has two solutions.

Q9. Simplify: (x³)² / x²

(A) x²(B) x³(C) x⁴(D) x⁵(E) x⁶

Show Answer

Answer: (C) x⁴
(x³)² = x⁶. Then x⁶ / x² = x^(6−2) = x⁴.

Q10. The product of two consecutive integers is 56. What are the integers?

(A) 6, 7(B) 7, 8(C) 8, 9(D) −7, −8(E) Both (B) and (D)

Show Answer

Answer: (E) Both (B) and (D)
Let the integers be n and n+1. Then n(n+1) = 56 → n² + n − 56 = 0 → (n−7)(n+8) = 0 → n = 7 or n = −8. So the pairs are (7, 8) or (−8, −7).

Q11. For what values of x is 2x + 3 > 5x − 9?

(A) x > 4(B) x < 4(C) x > −4(D) x < −4(E) x = 4

Show Answer

Answer: (B) x < 4
2x + 3 > 5x − 9 → 3 + 9 > 5x − 2x → 12 > 3x → 4 > x. We divided by +3, no flip needed.

Q12. If 5^(2k) = 125, what is k?

(A) 1/2(B) 3/2(C) 2(D) 3(E) 5

Show Answer

Answer: (B) 3/2
125 = 5³, so 5^(2k) = 5³ → 2k = 3 → k = 3/2. Always convert the target number to the same base.

Quick Reference Card

# ALGEBRA & EQUATIONS — KEY RULES

## Linear Equations

ax + b = c → x = (c − b) / a

## Systems (2 equations)

Add/subtract equations to eliminate one variable

OR substitute one equation into the other

## Quadratic Formula

x = (−b ± √(b²−4ac)) / 2a

disc > 0 → 2 real roots

disc = 0 → 1 real root (double)

disc < 0 → no real roots

## Inequalities

FLIP sign when × or ÷ by NEGATIVE

## Absolute Value

|x| = k → x = k OR x = −k (k ≥ 0)

|x| = negative → NO SOLUTION

|x| < k → −k < x < k

|x| > k → x > k OR x < −k

## Exponent Rules

a^m · a^n = a^(m+n)

a^m / a^n = a^(m−n)

(a^m)^n = a^(mn)

(ab)^n = a^n · b^n

a^0 = 1 (a ≠ 0)

a^(−n) = 1/a^n

a^(p/q) = q-th root of a^p

Hour 1: Number Properties

Hour 3: Word Problems