Foundations of Data Sufficiency: Algebra

Every Data Sufficiency question has the same five answer choices. You should memorize them so well that you do not have to read them on test day. They are:

• A: Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
• B: Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
• C: Both statements together are sufficient, but neither statement alone is sufficient.
• D: Each statement alone is sufficient.
• E: Statements (1) and (2) together are not sufficient.

Here is a simple flow to decide the letter:

• If (1) alone works and (2) alone works: choose D.
• If (1) alone works but (2) does not: choose A.
• If (2) alone works but (1) does not: choose B.
• If neither works alone, but together they work: choose C.
• If neither works alone and together they still do not work: choose E.

Notice that you never choose an answer before you have looked at both statements. Even if statement (1) looks perfect, you must still check statement (2) to see if it also works, because that changes the answer from A to D.

In algebra DS, the question usually gives you an equation or an inequality and asks you to find the value of a variable or to determine whether a condition holds. The statements provide additional equations, inequalities, or relationships. Your job is to decide whether the information is sufficient.

The core algebraic ideas are:

• Linear equations: equations of the form $ax + b = c$. They have one solution if $a \neq 0$.
• Systems of linear equations: two or more equations with the same variables. To solve for unique values of two unknowns, you generally need two independent linear equations.
• Quadratic equations: equations like $x^2 - 5x + 6 = 0$. They can have two solutions, one solution, or no real solutions.
• Inequalities: statements like $x > 3$ or $2 < x < 5$. They give ranges of possible values, not single values.

In DS, you rarely need to find the exact solution. You only need to know whether the solution is unique or whether the range is narrow enough to answer the question.

There are two main question types in DS: value questions and yes/no questions.

For example, if the question is “What is x?” and statement (1) says $x^2 = 4$, then x could be 2 or -2. That is not sufficient. But if the question is “Is x positive?” and statement (1) says $x^2 = 4$, then x could be positive or negative, so it is not sufficient either. However, if statement (1) says $x^3 = 8$, then x must be 2, so it is sufficient for both a value question and a yes/no question about positivity.

Use this three-step process for every DS question. It will keep you organized and prevent careless errors.

Step 1: Evaluate Statement (1) Alone

Ignore statement (2) completely. Ask: “Does statement (1) alone give me enough information to answer the question?” Write down your conclusion: sufficient or not sufficient.

Step 2: Evaluate Statement (2) Alone

Now ignore statement (1). Ask the same question about statement (2). Write down your conclusion.

Step 3: Combine If Neither Alone Is Sufficient

If both statements are insufficient alone, consider them together. Use the information from both as a single set of facts. Ask: “Together, do they give enough?” If yes, choose C. If no, choose E.

If one statement is sufficient alone, you never need to combine. You only need to check whether the other statement is also sufficient to decide between A, B, or D.

Algebra DS has several classic traps. Knowing them will save you time and points.

How Many Equations vs. How Many Unknowns

A single linear equation with two variables, such as $x + y = 5$, has infinitely many solutions. For example, x = 1, y = 4 or x = 2, y = 3. So it is not sufficient to find a unique value for x or y.

In general, to solve for n distinct variables, you usually need n independent linear equations. But this rule has exceptions. If the equations are not independent, such as $x + y = 5$ and $2x + 2y = 10$, they are essentially the same equation, so you still have only one equation. Also, if the question asks for a combination like x + y, sometimes one equation is enough.

Quadratic Solutions (Two Roots) Traps

A quadratic equation often has two solutions. For example, $x^2 - 5x + 6 = 0$ has solutions x = 2 and x = 3. If a statement gives such an equation, it is not sufficient for a value question asking for x, because there are two possible values.

But be careful: sometimes the question asks for something like x^2 or |x|, where two roots might give the same result. For instance, if $x^2 = 9$, then x = 3 or -3, but x^2 is always 9. So if the question asks for x^2, that statement is sufficient.

Inequalities and Ranges

An inequality like $x > 3$ gives a range of possible values, not a single value. So it is not sufficient for a value question. For a yes/no question, it might be sufficient if it forces a definite yes or no.

For example, “Is x > 0?” If statement (1) says $x > 5$, then x is definitely positive, so the answer is always yes. That is sufficient. If statement (1) says $x > -2$, then x could be -1 (no) or 0 (no) or 1 (yes), so it is not sufficient because the answer is sometimes yes and sometimes no.

Statement Interplay and Combining

When you combine statements, you are allowed to use both pieces of information together. Sometimes two insufficient statements become sufficient when combined. For example, statement (1) gives x + y = 10, and statement (2) gives x - y = 4. Each alone is insufficient, but together they form a system that solves to x = 7, y = 3.

However, combining does not always help. If the statements are contradictory, they are insufficient. For example, if statement (1) says x = 2 and statement (2) says x = 3, together they are contradictory, so no solution exists. In DS, if the statements are inconsistent, the answer is E (because together they do not provide a consistent answer).

Let’s practice with three examples that illustrate the traps.

Even at the sub-505 level, there are a few nuances that can surprise beginners.

Nonlinear Equations Can Have More Than Two Solutions

Equations like $x^3 = 8$ have one real solution (x=2), but equations like $x^4 = 16$ have two real solutions (x=2 and x=-2). In general, an even power gives two real roots, while an odd power gives one real root. This matters for sufficiency.

Combining Statements Can Contradict

If statement (1) says x = 2 and statement (2) says x = 5, then there is no value of x that satisfies both. In DS, that means the information is inconsistent, and the answer is E. Some students mistakenly choose C because they think “together they should work.” But if the statements conflict, they cannot be true simultaneously.

Beware of “Hidden” Equations

Sometimes a statement gives a relationship that is actually equivalent to another. For example, statement (2) might be the same as statement (1) multiplied by a constant. Recognizing this prevents you from thinking you have two independent equations when you only have one.

• Memorize the answer choices: A, B, C, D, E. Use the AD/BCE mnemonic to recall the structure.
• Always evaluate statement (1) alone, then statement (2) alone, then combine only if needed.
• For value questions, a statement is sufficient only if it yields exactly one value. Two roots or a range are not sufficient.
• For yes/no questions, a statement is sufficient if it gives a definite yes or a definite no. A maybe is not sufficient.
• One linear equation with two variables is not enough to solve for either variable. You generally need as many independent equations as unknowns.
• Quadratic equations can have two solutions, so they are often insufficient for value questions unless another condition narrows the choice.
• Inequalities give ranges, not exact values, so they are rarely sufficient for value questions but can be sufficient for yes/no questions.
• When combining, check for contradictions and redundancy. If the statements conflict or are the same, the answer is E.

With practice, you will develop the habit of thinking about sufficiency rather than solving. The goal is to answer the question with certainty, not to do unnecessary algebra. Keep this guide handy as you begin your DS practice.