Build exact algebraic habits across expressions, equations, inequalities, powers, distance, and proportional models—then retrieve them without the chapter label.
Name the variable, relation, unit, grouping, and permitted domain.
02Transform
Choose a valid symbolic move and state what it preserves or reverses.
03Verify
Substitute, test a region, or check sign, scale, base, and units.
04Decide
Return the requested expression, value, set, comparison, percent, or rate.
The five lesson labels follow the official Algebra scope. DAT TRAIN’s sequence, examples, and questions are original study tools and do not imply unpublished topic weights.
Five connected topic lessons
Keep a ledger for every claim.
Each model exposes the exact relation, transformation, boundary, comparison base, or unit that makes the conclusion valid.
01
LESSON 1 · 14 MIN
QR-MAT-EQU-01QR-MAT-EQU-02
Equations and expressions
Translate expressions, solve linear equations, and verify candidates while preserving grouping, signs, and restrictions.
ESSENTIAL QUESTIONWhich transformations preserve the original relationship—and how can substitution prove it?
SEMANTIC MATH MODEL · COMPLETE TEXT AND TABLE EQUIVALENT INCLUDED
01
Translate the relationship
Build noun phrases and relational phrases before following the surface order of the words.
‘Less than’ reverses the written order.
A fraction bar groups its entire numerator and denominator.
02
Preserve equivalence
Apply an invertible operation to both sides and record any restriction introduced by a conditional move.
Name the operation before performing it.
Do not cancel terms across addition.
03
Verify in the original
Substitute the candidate into the original equation rather than a transformed line.
Compare both sides exactly.
Reject excluded values and extraneous candidates.
Worked example
Solve 5(x − 2) + 3 = 18 and verify the result.
1
Expand or first subtract 3: 5(x − 2) = 15.
2
Divide by 5: x − 2 = 3, so x = 5.
3
Original check: 5(5 − 2) + 3 = 15 + 3 = 18.
Conclusionx = 5 is the unique solution because the reversible steps preserve equality and the original check succeeds.
Calculator decision ledger
Model mentally
Translate and isolate symbolically.
Estimate
Predict sign and approximate magnitude.
Calculator
Use only for cumbersome arithmetic.
Audit
Substitute into the original equation.
Close the notes first
Retrieve the model.
01Translate ‘four more than three times y.’
3y + 4
‘Three times y’ is the base quantity; four is added to it.
02Solve 4x + 7 = 31.
x = 6
Subtract 7, divide by 4, then verify 24 + 7 = 31.
03Why check a candidate in the original equation?
Conditional transformations can create or retain invalid candidates.
The original relation is the actual acceptance condition.
02
LESSON 2 · 14 MIN
QR-MAT-INE-01QR-MAT-INE-02
Inequalities
Solve linear, compound, and absolute-value inequalities and express the complete region with correct direction and endpoints.
ESSENTIAL QUESTIONWhen does an algebraic move change order, and what does the resulting region include?
Semantic inequality modelDirection, boundary, and region stay explicit.
−3x + 2 > 11→x < −3
Text equivalent: The boundary is negative three and is excluded. Every number less than negative three is included; every number greater than or equal to negative three is excluded.
−7−5−3−11
Open boundary at −3; solution extends left
Inequality transformation and endpoint ledger
Pass
Statement
Reason
Subtract
−3x > 9
Adding the same value preserves order
Divide
x < −3
Division by −3 reverses order
Endpoint
Open at −3
The inequality is strict
Region test
x = −4 works
14 > 11 in the original inequality
SEMANTIC MATH MODEL · COMPLETE TEXT AND TABLE EQUIVALENT INCLUDED
01
Protect direction
Adding preserves order; multiplying or dividing by a negative reverses it.
Record the sign of the divisor.
Reverse once—not once per negative term.
02
Classify endpoints
Strict inequalities exclude a boundary; inclusive inequalities include it.
Open means excluded.
Closed means included.
03
Combine regions
An ‘and’ condition requires intersection; an ‘or’ condition permits union.
Test a point in each region.
State the whole set, not one example.
Worked example
Solve −3x + 2 > 11 and describe the number line.
1
Subtract 2: −3x > 9.
2
Divide by −3 and reverse the sign: x < −3.
3
Use an open endpoint at −3 and include all values to its left.
ConclusionThe solution is x < −3, or (−∞, −3); −3 itself is excluded.
Calculator decision ledger
Model mentally
Mark direction, endpoint type, and set operation.
Estimate
Test one value on each side of a boundary.
Calculator
Usually unnecessary for foundational coefficients.
Audit
Verify a boundary and representative region value.
Close the notes first
Retrieve the model.
01What happens to < after division by −5?
It reverses to >.
Multiplication or division by a negative reverses order.
02Write −2 ≤ x < 4 in interval notation.
[−2, 4)
The left endpoint is included; the right endpoint is excluded.
03What set operation usually matches ‘and’?
Intersection
A value must satisfy both conditions simultaneously.
03
LESSON 3 · 12 MIN
QR-MAT-EXP-01
Exponential notation
Apply exponent and scientific-notation rules without losing the base, grouping, reciprocal, or order of magnitude.
ESSENTIAL QUESTIONWhat exactly is the base, and what does the exponent act on?
Exponent and magnitude ledgerName the base before applying a rule.
3² · 3⁻⁵→1/27
Exponent structure, exact value, and magnitude checks
Expression
Structure
Exact result
Magnitude audit
2⁻³
Negative exponent
1/8
Positive and below 1
3² · 3⁻⁵
Same-base product
3⁻³ = 1/27
Positive and below 1
0.00045
Decimal moved 4 places right
4.5 × 10⁻⁴
Coefficient is in [1, 10)
(−2)⁴ versus −2⁴
Grouped versus ungrouped sign
16 versus −16
Parentheses change the base
SEMANTIC MATH MODEL · COMPLETE TEXT AND TABLE EQUIVALENT INCLUDED
01
Name the base
Parentheses determine whether a sign or product belongs to the powered quantity.
(−2)⁴ and −2⁴ differ.
An exponent applies before an ungrouped leading sign.
02
Combine only like bases
Products of equal bases add exponents; powers of powers multiply exponents.
Do not combine exponents across addition.
Keep the reciprocal exact.
03
Audit magnitude
Scientific notation uses a coefficient with absolute value from 1 up to, but not including, 10.
Small positive decimals use negative exponents.
Move and exponent must offset each other.
Worked example
Simplify (2³ · 2⁻⁵) and estimate its size.
1
The bases match, so add exponents: 3 + (−5) = −2.
2
Rewrite 2⁻² as the reciprocal 1/2².
3
Evaluate exactly: 1/4, which is positive and less than 1 as expected.
ConclusionThe exact value is 1/4; the negative exponent creates a reciprocal, not a negative result.
Calculator decision ledger
Model mentally
Identify base and exponent operation.
Estimate
Predict sign and whether magnitude is above or below 1.
Calculator
Confirm a decimal only after exact simplification.
Audit
Check coefficient range and decimal direction.
Close the notes first
Retrieve the model.
01Rewrite 5⁻² without a negative exponent.
1/25
A negative exponent takes the reciprocal of the positive power.
02Simplify a⁴a⁻¹ for a ≠ 0.
a³
Equal bases in a product add exponents.
03Write 0.0062 in scientific notation.
6.2 × 10⁻³
The decimal moves three places right, so the power of ten is −3.
04
LESSON 4 · 12 MIN
QR-MAT-ABS-01
Absolute value
Interpret absolute value as distance, generate all valid cases, and reject impossible negative-distance claims.
ESSENTIAL QUESTIONWhich points are the stated distance from the reference point?
Absolute-value distance modelTwo directions can share one distance.
|2x − 6| = 10→x = −2 or 8
Text equivalent: The two solution points are negative two and eight. Their midpoint is three, and each is five units from three.
−20368
Two solutions symmetric around x = 3
Absolute-value case and substitution ledger
Case
Equation
Candidate
Original check
Positive direction
2x − 6 = 10
x = 8
|16 − 6| = 10
Negative direction
2x − 6 = −10
x = −2
|−4 − 6| = 10
Feasibility
|u| = −2
None
Distance cannot be negative
SEMANTIC MATH MODEL · COMPLETE TEXT AND TABLE EQUIVALENT INCLUDED
01
Read distance
|x − c| measures the distance from x to c on the number line.
Distance is never negative.
The reference point is c, not −c.
02
Build cases
A positive target distance generates a point on each side of the center.
Solve the positive case.
Solve the negative case.
03
Check symmetry
Valid solutions lie equally far from the center unless a surrounding restriction removes one.
Count all candidates.
Test each in the original bars.
Worked example
Solve |2x − 6| = 10.
1
Set 2x − 6 = 10, giving x = 8.
2
Set 2x − 6 = −10, giving x = −2.
3
Check: both inside values have magnitude 10.
ConclusionThe solution set is {−2, 8}; keeping only one case loses half the distance model.
Calculator decision ledger
Model mentally
Identify center and target distance.
Estimate
Predict two symmetric cases, one case, or none.
Calculator
Use only if case arithmetic is cumbersome.
Audit
Substitute every candidate into the original bars.
Close the notes first
Retrieve the model.
01What is |−9|?
9
−9 is nine units from zero.
02Solve |x| = 3.
x = −3 or x = 3
Both points are three units from zero.
03Can |u| = −1 have a real solution?
No
A distance cannot be negative.
05
LESSON 5 · 14 MIN
QR-MAT-RAT-01QR-MAT-RAT-02
Ratios and proportions
Align comparison bases and units, then distinguish direct, inverse, percent-change, and unit-rate models.
ESSENTIAL QUESTIONWhich ratio or product must stay invariant as the quantities change?
Ratio and unit ledgerChoose the invariant before the arithmetic.
6 workers · 10 days = 15 workers · d days→d = 4 days
Proportion type, invariant, unit, and direction ledger
Model
Invariant
Unit result
Direction check
Direct proportion
Equal ordered ratios
Same numerator unit
Both quantities scale together
Inverse work model
Workers × days
Days
More workers, fewer days
Percent change
Change ÷ original
Percent
Sign matches increase or decrease
Unit rate
Quantity ÷ elapsed unit
Miles per hour
Scale returns 180 miles in 3 hours
SEMANTIC MATH MODEL · COMPLETE TEXT AND TABLE EQUIVALENT INCLUDED
01
Align the base
A ratio is ordered; both sides of a comparison must describe the same quantities in the same units.
Label numerator and denominator.
Convert units before comparing.
02
Choose direction
Direct proportionality keeps a ratio constant; inverse proportionality keeps a product constant.
Ask whether outputs should rise or fall together.
Use the verbal relationship, not a keyword alone.
03
Audit scale
Percent change ordinarily uses the original value, while a unit rate divides by one unit of the denominator.
Name the comparison base.
Attach the resulting unit.
Worked example
Six workers need 10 days. At the same rate, how many days would 15 workers need?
1
More workers should require fewer days, so use an inverse model.
2
Preserve worker-days: 6 workers × 10 days = 15 workers × d days.
3
Solve d = 60/15 = 4 and check that the direction is reasonable.
ConclusionFifteen workers need 4 days under the stated equal-rate assumption.
Calculator decision ledger
Model mentally
Label quantities, units, and direct or inverse direction.
Estimate
Predict whether the answer grows or shrinks.
Calculator
Perform the final division when useful.
Audit
Check units, comparison base, and direction of change.
Close the notes first
Retrieve the model.
01Solve 3/5 = x/20.
x = 12
The denominator scales by four, so the numerator does too.
02What base is used for ordinary percent increase?
The original value
Change is compared with the starting quantity.
03What stays constant in a simple inverse proportion?
The product of the paired quantities
One quantity decreases in the same factor that the other increases.
Randomized retrieval bank
Now solve without the lesson label.
All eight objectives appear in the bank. Every question and all five choices shuffle while each answer remains attached to its independently recomputed key.
20 PRACTICE QUESTIONS
Retrieve before you review.
Question order and all five answer options are shuffled when you begin. The correct answer stays attached to the same underlying choice.
Scope and score notice
Use results for study guidance.
The ADA names these Algebra topics but does not publish their individual question quotas. DAT TRAIN does not invent them.
These original questions are draft and uncalibrated. Accuracy can direct review; it cannot predict an official DAT score.