ASVAB Algebra: Equations, Exponents and Factoring

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An equation is a relationship to preserve

ASVAB algebra practice should begin with one reliable habit: perform the same valid operation on both sides of an equation. The equal sign says the expressions have the same value. Your goal is to isolate the unknown while preserving that equality, not simply move symbols around until the answer looks familiar.

For 3x + 7 = 22, subtract seven from both sides to get 3x = 15, then divide both sides by three to get x = 5. Substitute five into the original: 3 × 5 + 7 = 22. That final check can catch a sign error that is difficult to see in your intermediate lines.

If fractions and negative numbers are shaky, review them alongside equations. Algebra often exposes an arithmetic gap rather than an entirely new concept.

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Distribute before collecting terms

Solve 2(x + 3) = 18. Distribute the two to both terms: 2x + 6 = 18. Subtract six, giving 2x = 12, then divide by two: x = 6. You could also divide the original equation by two first, producing x + 3 = 9. Both paths preserve equality.

A negative sign affects every term inside parentheses. For 10 − (x + 4), the equivalent expression is 10 − x − 4 = 6 − x. Writing 10 − x + 4 changes the value. Treat subtraction of a group as multiplication of that group by negative one.

Combine like terms only. Three x terms plus two x terms equals five x terms. Three x terms plus two constants remains 3x + 2. Terms involving x and x² are not like terms, because they represent different powers. Say what each term means before combining it.

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Variables on both sides and inequalities

For 5x − 4 = 2x + 11, subtract 2x from both sides to obtain 3x − 4 = 11. Add four, then divide by three: x = 5. Substitution gives 21 on both sides. Write one operation per line when learning so you can identify the exact point of an error.

Inequalities follow similar operations with one extra rule: multiplying or dividing both sides by a negative number reverses the inequality direction. For −2x < 8, division by −2 gives x > −4. Check a value such as zero: −2 × 0 = 0, and zero is less than eight, so zero belongs in the solution.

Do not reverse an inequality merely because you subtract a number. The reversal concerns multiplication or division by a negative quantity. A number line can help you check whether your final statement describes values above or below a boundary.

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Exponents and factoring with examples

For the same nonzero base, multiplication adds exponents: x² × x³ = x⁵. Division subtracts exponents: x⁵ ÷ x² = x³, with x not equal to zero. Raising a power to a power multiplies exponents: (x²)³ = x⁶. Addition is different: x² + x³ does not become x⁵.

Factor 6x + 12 by taking out the common factor six: 6(x + 2). Factoring reverses distribution. Check by multiplying back out. For x² + 5x + 6, look for two numbers that multiply to six and add to five: two and three. The expression factors as (x + 2)(x + 3).

If (x + 2)(x + 3) = 0, at least one factor must be zero, so x = −2 or x = −3. This reasoning applies to a product equal to zero. It does not justify setting each factor to zero when the product equals a different number.

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Choose practice that tests the decision

Rotate one-step equations, parentheses, variables on both sides, inequalities, and simple factoring. Before solving, name the first useful operation. After solving, verify in the original statement. A check that uses your already-simplified but incorrect line can preserve the same mistake.

When a problem provides answer choices, substitution can sometimes be efficient. It is still reasoning: you are checking which candidate satisfies the equation. Avoid testing choices blindly if the equation can be solved in two short steps. Compare methods during study and choose the clearer one.