Algebra Review Test

Review essential Algebra 1 and Algebra 2 skills in one balanced 20-question test.

Algebra Review Test

20 mixed algebra questions with instant feedback and worked explanations.

Instant feedback · Worked explanations

Algebra Review Compass

A useful review score starts with knowing which rule belongs to which structure.

A cumulative Algebra 1 and early Algebra 2 review should feel different from a single-topic drill. The challenge is recognition: simplify an expression without mixing unlike terms, solve an inequality with the correct sign logic, preserve restrictions in a rational expression, identify a function rule, distinguish factoring from discriminant reasoning, and translate a short area model into algebra. Each result becomes more reliable when a quick substitution or structural check is possible.

Like termsDistributionLinear equationsInequalitiesAbsolute valueFactoringQuadraticsSystemsExponent lawsRational expressionsRadicalsFunctionsSequencesLinesArea models

1. Symbolic manipulation starts with matching structure

Like terms combine only when their variable parts match, and distribution reaches every term inside a group.

Combine like terms

4 x + 3 - 2 x + 5

2 x + 8

Variable terms combine with matching variable terms, while constants combine separately.

Distribute before combining

3 (x - 2) + 2 x

3 x - 6 + 2 x

5 x - 6

The outside multiplier applies to every term inside the parentheses.

2. Equations and inequalities preserve balance in different ways

Equations keep equality balanced; inequalities add one extra rule when a negative factor changes order.

Linear equation

5 x - 7 = 3 x + 9

2 x = 16

x = 8

40 - 7 = 24 + 9

Substitution confirms that both sides agree.

Linear inequality

- 2 x + 5 > 11

- 2 x > 6

x < - 3

The symbol reverses because the final division is by a negative number.

3. Absolute value creates branches instead of one ordinary linear path

A positive absolute-value target represents the same distance on two sides of a center.

Start

| x - 4 | = 6

Interpret the absolute value as a distance.

Split into two equations

x - 4 = 6

x - 4 = - 6

Keep both valid branches

x = 10, x = - 2

Each solution gives the required distance.

4. Factoring and discriminants answer different quadratic questions

Factoring exposes roots directly; the discriminant classifies how many real roots a quadratic has.

Factored roots

x^{2} - 5 x + 6

(x - 2) (x - 3)

x = 2, x = 3

Repeated root from the discriminant

x^{2} - 4 x + 4 = 0

D = 16 - 16

D = 0

x = 2

A zero discriminant signals one repeated real root.

5. A system solution must satisfy both equations simultaneously

Elimination is efficient when adding the equations removes one variable immediately.

Eliminate one variable

x + y = 7

x - y = 1

2 x = 8

x = 4

Recover the second coordinate

y = 3

(4, 3)

The ordered pair is accepted only because it satisfies both original equations.

6. Exponent laws and rational expressions both need restrictions before cancellation

Matching bases justify exponent rules; original denominators preserve excluded values.

Exponent structure

\frac{x^{3} \cdot x^{2}}{x}

\frac{x^{5}}{x}

x \neq 0

x^{4}

The denominator requires a nonzero input before cancellation.

Rational simplification

\frac{x^{2} - 9}{x^{2} - x - 6}

\frac{(x - 3) (x + 3)}{(x - 3) (x + 2)}

x \neq 3, x \neq - 2

\frac{x + 3}{x + 2}

Cancel factors, not arbitrary terms, and keep every original exclusion.

7. Radicals and functions both reward direct substitution checks

Radical conditions protect the domain, while function evaluation replaces the input everywhere in the rule.

Radical equation

\sqrt{x + 5} = 4

x \geq - 5

x + 5 = 16

x = 11

Function evaluation

f (x) = 2 x^{2} - 3

f (- 2) = 2 {(- 2)}^{2} - 3

f (- 2) = 5

The complete input is substituted into every occurrence of the variable.

8. Arithmetic sequences encode repeated additive change

The common difference is multiplied by one less than the term number.

General rule

a_{n} = a_{1} + (n - 1) d

Known values

a_{1} = 5, d = 3

a_{6} = 5 + 5 \cdot 3

Requested term

a_{6} = 20

The sequence grows by the same additive amount each step.

9. Line questions connect points, slope, and intercepts

The coordinate graph makes the rise-over-run interpretation visible before the equation is written.

From two points to a line

(1, 3)

(5, 11)

m = \frac{11 - 3}{5 - 1}

m = 2

y = 2 x + 1

b = 1

The slope measures vertical change per horizontal change; the constant gives the vertical intercept.

10. Area models turn geometry into a quadratic equation

The diagram helps distinguish side lengths from the product that represents area.

Translate dimensions into an area equation

x + 2

x + 5

(x + 2) (x + 5) = 70

x^{2} + 7 x - 60 = 0

(x + 12) (x - 5) = 0

x = 5

The negative algebraic root is rejected because a geometric side length must be positive.

11. Diagnostic errors usually point directly to the missing Algebra skill

A cumulative review becomes useful when the mistake is classified, not just marked wrong.

Unlike terms combined

$3 x + 2 y$ contains different variable parts, so those terms remain separate.

Inequality direction not reversed

$x < - 3$ appears only after dividing the previous inequality by a negative number.

Absolute-value branch lost

$x = 10, x = - 2$ shows the two valid branches.

Terms canceled instead of factors

$x \neq 3, x \neq - 2$ remains part of the rational expression’s domain history.

Exponent law applied to the wrong structure

$x^{3} \cdot x^{2} = x^{5}$ works because the bases match.

Familiar rule used on the wrong problem family

Classify the expression before choosing a procedure.

Final Algebra Review audit

Before accepting an answer, confirm the problem family, sign logic, restrictions, branches, and one quick verification step.

What algebra structure is this?Expression, equation, inequality, polynomial, system, function, sequence, or model?

Are equivalent operations being used correctly?Preserve equality and watch negative division in inequalities.

Are restrictions or multiple branches present?Keep rational exclusions, radical conditions, and absolute-value branches visible.

Does the chosen rule match the exact expression type?Exponent, factoring, and discriminant rules are structural, not interchangeable.

Can a graph or model confirm the interpretation?Slope and area questions benefit from a visual structural check.

Can I substitute the result back quickly?A short check catches many sign, domain, and arithmetic errors.

This block supports the Algebra Review Test. Its examples are illustrative rather than copies of the test questions. The focus is combining like terms, distribution, linear equations and inequalities, absolute value, polynomial factorization, quadratic roots and discriminants, systems of equations, exponent laws, rational expressions, radicals, function evaluation, arithmetic sequences, lines, and area models.