Mixed Algebra Practice Test

Review a balanced mix of Algebra 1 and Algebra 2 skills in one diagnostic practice test.

Mixed Algebra Practice Test

20 mixed algebra questions with immediate explanations and incorrect-choice analysis.

Instant feedback · Worked explanations

Mixed Algebra Diagnostic Mosaic

A mixed test measures whether you can choose the method before using it.

This review blends Algebra 1 and Algebra 2 skills so the first step is recognition. A distribution problem, inequality, system, rational expression, radical equation, logarithm, inverse function, or sequence may look similar at first glance, but each has its own structural rules. The strongest approach is to classify the problem, preserve restrictions, solve with the appropriate method, and then verify any candidate that may be excluded or extraneous.

DistributionLinear equationsInequalitiesAbsolute valueSystemsFunctionsFactoringQuadraticsRational expressionsRadicalsExponentsInverse functionsSequencesCoordinate geometryLogarithmsWord problems

1. Distribution and linear equations reward careful sign control

Expand every grouped term first, then combine like terms or preserve equality as needed.

Simplify an expression

2 (3 x - 4) - 5 (x + 1)

6 x - 8 - 5 x - 5

x - 13

Every term inside each parenthesis receives its outside multiplier.

Solve a multi-step equation

4 (x - 2) + 3 = 2 x + 11

4 x - 5 = 2 x + 11

2 x = 16

x = 8

After simplifying both sides, preserve equality while isolating the variable.

2. Inequalities and absolute value both change the usual one-path solving pattern

A negative division reverses inequality order, while a positive absolute-value equation splits into two cases.

Reverse only at the negative division step

- 3 (2 x - 1) > 9

- 6 x + 3 > 9

- 6 x > 6

x < - 1

Split an absolute-value equation

| 2 x - 5 | = 7

2 x - 5 = 7

2 x - 5 = - 7

x = 6, x = - 1

Both branches must be solved because both have the required distance from the center.

3. Systems and functions both use substitution, but for different purposes

A system uses substitution to satisfy two relations simultaneously; a function uses substitution to evaluate one rule at a chosen input.

System by substitution

y = x + 1

2 x + y = 10

3 x + 1 = 10

x = 3

y = 4

(3, 4)

Function simplification with a restriction

f (x) = \frac{x^{2} - 1}{x - 1}

x \neq 1

f (x) = x + 1

f (3) = 4

The canceled factor does not restore the excluded input.

4. Factoring and square-root solving expose different kinds of roots

Factored quadratics use the zero-product property; squared equations require both square-root branches.

Factor a quadratic

x^{2} - 5 x + 6

(x - 2) (x - 3)

x = 2, x = 3

Keep the negative square-root branch

x^{2} = 18

x = \pm \sqrt{18}

x = \pm 3 \sqrt{2}

Both signs square back to the same positive value.

5. Rational expressions keep every restriction from the original denominator

Factor before canceling, and record excluded values before simplifying.

Original expression

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

Find all denominator values that make the original expression undefined.

Factor completely

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

x \neq 3, x \neq - 2

Restrictions belong to the original form.

Simplify

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

The canceled restriction still remains excluded.

6. Radical equations need a check after squaring

Squaring preserves every true solution, but it can also create a candidate that does not satisfy the original radical equation.

Create the polynomial candidates

\sqrt{x + 5} = x - 1

x \geq 1

x + 5 = {(x - 1)}^{2}

x^{2} - 3 x - 4 = 0

(x - 4) (x + 1) = 0

Check the candidates

x = 4

x = - 1

The negative candidate fails the original equation and must be rejected.

7. Exponent laws and logarithms both depend on structural restrictions

Exponent rules require matching bases, while logarithms require a positive argument.

Exponent simplification

\frac{{(x^{3})}^{2}}{x^{4}}

\frac{x^{6}}{x^{4}}

x \neq 0

x^{2}

The denominator creates a nonzero restriction before cancellation.

Logarithmic equation

\log_{2} (x - 1) = 3

x > 1

x - 1 = 8

x = 9

The final candidate satisfies the required positive logarithm argument.

8. Inverse functions and sequences look different because they reverse different structures

An inverse function reverses input and output; a geometric sequence applies repeated multiplication.

Find an inverse function

f (x) = 2 x + 5

y = 2 x + 5

x = 2 y + 5

y = \frac{x - 5}{2}

f^{- 1} (x) = \frac{x - 5}{2}

Geometric sequence

a_{n} = 5 \cdot 3^{n - 1}

a_{4} = 5 \cdot 3^{3}

a_{4} = 135

The common ratio is multiplied repeatedly rather than added repeatedly.

9. Coordinate geometry links points, slope, and a line equation

Slope is vertical change divided by horizontal change, and a line equation must fit both points.

(- 2, 3)

(4, 15)

m = \frac{15 - 3}{4 - (- 2)}

m = 2

y = 2 x + 7

Substituting either point into the line equation verifies the relationship.

10. Word problems start with the model, not the arithmetic

Define what the variable represents, translate the relationship, and then solve.

Build the equation

18 + 7 x = 67

A fixed fee plus a per-use amount equals the total.

Remove the fixed part

7 x = 49

Preserve equality while isolating the variable term.

Interpret the result

x = 7

The numerical solution is read back in the context of the problem.

11. Expansion and substitution can verify unfamiliar answer forms

When two forms do not look alike, expand or substitute instead of guessing whether they are equivalent.

Original form

{(x + 2)}^{2} - {(x - 2)}^{2}

The expression contains two squared binomials.

Expand both

(x^{2} + 4 x + 4) - (x^{2} - 4 x + 4)

Matching terms reveal what cancels.

Simplified result

8 x

Expansion confirms the unfamiliar form rather than relying on appearance.

12. Diagnostic errors usually reveal the topic that was misidentified

A mixed test is valuable because the mistake often points directly to the missing rule.

Distribution sign error

Carry every positive or negative multiplier through every term in the parentheses.

Inequality not reversed

$x < - 1$ appears only after dividing the previous inequality by a negative number.

Negative quadratic root omitted

$x = \pm 3 \sqrt{2}$ keeps both square-root branches.

Rational restriction lost

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

Exponent rule used on unlike structure

$x^{3} \cdot x^{4} = x^{7}$ works because the bases match.

Stopped before the requested result

$8 x$ is the fully simplified form after expansion and cancellation.

Final mixed-algebra audit

Before accepting an answer, confirm the topic, restrictions, sign logic, candidate checks, and whether the requested form is complete.

What algebra family is this?Identify the topic before choosing a procedure.

What restrictions exist before simplifying?Preserve rational, logarithmic, and exponent-domain conditions.

Does the sign logic change the solution?Watch distribution and negative division in inequalities.

Could the process create or hide a candidate?Check radical candidates and both quadratic square-root branches.

Can substitution or expansion verify the answer form?Use a direct check when forms look unfamiliar.

Did I finish in the form the question requests?Do not stop one algebraic step before the actual target.

This block supports the Mixed Algebra Practice Test. Its examples are illustrative rather than copies of the test questions. The focus is distribution, linear equations, inequalities, absolute value, systems, function evaluation, polynomial factoring, quadratics, rational restrictions, radical equations, exponent laws, inverse functions, geometric sequences, coordinate geometry, logarithmic equations, word problems, and verification by substitution or expansion.