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ALEKS placement math guide

ALEKS math placement example questions with worked steps

These are example questions for the kinds of topics the ALEKS placement math test uses, each solved one line at a time. They are written for Lemma, not copied from ALEKS. Use them to see what a full answer looks like, then try a fresh one on the topic page.

What an ALEKS placement question looks like

ALEKS placement is not multiple choice. At the University of Rhode Island, the assessment is described as open-response: you work the problem out on paper, then type the answer in. It is adaptive, so the next question depends on how you did on the last one, and once you submit an answer you cannot change it.

A calculator is not allowed unless ALEKS shows one on screen for that problem. That means you need to be comfortable with fractions, signs and exponents by hand. If you do not know how to start a question, the same school advises answering I do not know, which tells ALEKS what to teach you next.

Lemma is independent practice and is not affiliated with ALEKS or McGraw Hill. These examples show the type of work, not the questions you will get.

Equations and inequalities

Equation questions ask you to find the value of a variable. Show each move so a slip is easy to find. Variables on both sides is a common place to lose a sign.

Variables on both sides

Worked example

Solve 5x - 3 = 2x + 12.

  1. 3x−3=12Subtract 2x from both sides.
  2. 3x=15Add 3 to both sides.
  3. x=5Divide by 3.

More both sides practice

Absolute-value equations

Worked example

Solve 2|x - 3| + 1 = 11.

  1. ∣x−3∣=5Subtract 1, then divide by 2 to isolate the bars.
  2. x−3=5orx−3=−5The inside can be 5 units away in either direction.
  3. x=8orx=−2Solve each branch.

More absolute value practice

Linear inequalities

Worked example

Solve -3x + 5 > 14.

  1. −3x>9Subtract 5 from both sides; the symbol does not change.
  2. x<−3Dividing by -3 reverses the inequality.

More inequalities practice

Lines and graphs

Questions like "find the slope of the line graphed below" or "find an equation for the line below" give you a graph or two points. Read two exact points on the line, then use rise over run. Once you have the slope, point-slope form builds the equation.

Slope from two points

Worked example

Find the slope through (1, -2) and (5, 6).

  1. m=5−16−(−2)​Second point minus first, in both coordinates.
  2. m=48​=2Rise 8 over run 4 reduces to 2.

More slope practice

Equations of lines

Worked example

Write the line through (2, -1) with slope 3.

  1. y−(−1)=3(x−2)Point-slope with (x₁, y₁) = (2, -1).
  2. y+1=3x−6Simplify and distribute.
  3. y=3x−7Solve for y.

More line equations practice

Read linear graphs

Worked example

A line crosses the y-axis at 4 and passes through (3, -2). Write its equation.

  1. m=3−0−2−4​Use the intercept (0, 4) as the first point and (3, -2) as the second. Subtract in that order on top and bottom.
  2. m=−2-6 over 3 is -2, so the line falls.
  3. y=−2x+4Substitute m and b into slope-intercept form.

More linear graphs practice

Factoring, quadratics and rational expressions

Factor quadratic trinomials

Worked example

Factor x² − 7x + 12

  1. (−3)(−4)=12The constants in the two groups must multiply to twelve. Two negative numbers give a positive product.
  2. (−3)+(−4)=−7They must also add to negative seven, the coefficient of the middle term. This pair passes both checks.
  3. x2−7x+12=(x−3)(x−4)Use the pair to build two factors. Factoring reverses multiplication.
  4. (x−3)(x−4)=x2−4x−3x+12Check by multiplying each term in the first group by each term in the second.
  5. x2−4x−3x+12=x2−7x+12The two middle products combine to the original middle term. All three terms match.

More factor trinomials practice

Solve quadratics by factoring

Worked example

Solve x² + 3x = 10

  1. x2+3x−10=0The zero-product rule needs a zero on one side.
  2. (x+5)(x−2)=0Five and negative two multiply to negative ten and add to three. They give the two factors.
  3. x+5=0orx−2=0At least one factor must be zero for their product to be zero. Each equation is a separate possibility.
  4. x=−5orx=2Solve each small equation. Either value makes the original left side equal ten.

More solve by factoring practice

Rational equations

Worked example

Solve 1/x + 1/2 = 3/(2x)

  1. x=0A zero input would make an original denominator zero. Exclude it before multiplying.
  2. 2x⋅x1​+2x⋅21​=2x⋅2x3​The least common denominator is twice the variable. Multiply every term on both sides by it.
  3. 2+x=3Cancel within each product. This step preserves the equation on the original allowed domain.
  4. x=1Subtract two from both sides. One is a candidate solution and is not excluded.
  5. 11​+21​=2(1)3​Substitute the candidate into the original equation, where the denominators still matter.
  6. 23​=23​Both sides have the same value. The allowed candidate is therefore a solution.

More rational equations practice

Functions, logarithms and sequences

In Lemma's plan, logarithms and sequences sit in the higher checkpoints. Evaluating a log means finding the exponent. A sequence question usually gives a few terms and asks for a later term or a sum.

Evaluate logarithms

Worked example

Evaluate log₄(64).

  1. 4y=64Convert to exponential form.
  2. y=34³ = 64, so the exponent is 3.

More evaluate logs practice

Arithmetic and geometric sequences

Worked example

Find the 12th term of 5, 8, 11, 14, …

  1. d=3Each term adds 3.
  2. a12​=5+(12−1)(3)Substitute into the arithmetic formula.
  3. a12​=38Eleven gaps of 3 is 33, plus the first term 5.

More sequences practice

How to use these examples

  1. Cover the steps and try the example first on paper.
  2. Open the steps and compare line by line. Find the first line where yours differs.
  3. Go to the topic page and work a fresh problem. Lemma checks each line you type.

More topics to practice

Practice it on Lemma

Type each step and Lemma checks it. Free, no account, no email.

See every ALEKS placement math topic

Questions

Are these the real ALEKS placement questions?

No. ALEKS does not publish its question bank. These are original examples of the same kinds of topics, written for Lemma.

Is the ALEKS placement test multiple choice?

No. One school's FAQ describes it as open-response: you work the problem on paper and type the answer. Your own school's page has the details for your test.

Can I use a calculator on the ALEKS placement test?

Only when ALEKS puts an on-screen calculator on a problem. Otherwise it is not allowed at the school quoted here, so practice fractions and signs by hand.

What if I do not know how to start a question?

The University of Rhode Island says answering I do not know is appropriate for material you have not learned, and that a wrong answer carries no penalty. It helps ALEKS see what you know.

More guides

Sources

Facts checked 2026-10-14. Independent practice, not affiliated with ATI or McGraw Hill.