QUANTITATIVE REASONING · ALGEBRA FOUNDATIONS

Preserve the structure.
Then solve.

Eight study objectives connect five official algebra scope lines. Move from translation and equivalence to regions, powers, distance, and proportional invariants.

Start five guided lessons →Build a balanced set ↗See every objective ↓
5official topics
8study objectives
5guided lessons
0invented weights

Official scope

Five algebra lines.
No fabricated quotas.

The ADA lists Equations and expressions, Inequalities, Exponential notation, Absolute value, and Ratios and proportions within the Algebra scope. Graphical analysis remains a separate learning route.

The eight DAT TRAIN objectives organize instruction. They do not predict ADA question counts, topic weights, order, or difficulty. This route is a learning bank, not a calibrated form or score converter.

OFFICIAL HIERARCHY → STUDY SEQUENCE

Five branches. Eight outcomes.

Open a branch to inspect the outcome, evidence boundary, known trap, representations, and free source trail before beginning its lesson.

A1Equations and expressions2 objectives
Objective 1QR-MAT-EQU-01

Translate structure before simplifying

Convert verbal or nested algebraic relationships into an expression or equation while preserving grouping, sign, and unit meaning.

Must know
Operation order follows the written structure, including grouping symbols and fraction bars. A variable expression represents a relationship; an equation adds an equality constraint.
Evidence boundary
Do not infer a missing domain restriction, unit, or relationship that the prompt does not supply.
Common trap
Words can be translated in the order they appear. Relational phrases determine grouping and operation direction before arithmetic begins.
Representations
text · equation · unit ledger
Objective 2QR-MAT-EQU-02

Solve and verify equations

Solve linear, quadratic, rational, or simultaneous equations at the stated level and verify candidates in the original constraints.

Must know
Equivalent transformations preserve the solution set only when their conditions are respected. Substitution into the original equation catches extraneous or excluded values.
Evidence boundary
No advanced mathematics or calculus; choose methods supported by the supplied algebraic structure.
Common trap
Every value produced by an algebraic manipulation is a solution. Squaring, clearing denominators, and other conditional moves can create candidates that fail the original equation.
Representations
equation · table
A2Inequalities2 objectives
Objective 1QR-MAT-INE-01

Preserve order through transformations

Solve one-variable inequalities and express the complete solution set using symbols, intervals, or a number line.

Must know
Multiplying or dividing by a negative reverses the inequality direction. Strict and inclusive boundaries produce different endpoint notation.
Evidence boundary
Report the full solution region rather than selecting a convenient test value as the answer.
Common trap
An inequality sign behaves exactly like an equals sign under every operation. Order reverses under multiplication or division by a negative quantity.
Representations
inequality · number line
Objective 2QR-MAT-INE-02

Resolve compound and absolute inequalities

Combine intersection or union conditions and test boundary regions for compound, rational, or absolute-value inequalities.

Must know
‘And’ usually describes an intersection; ‘or’ usually describes a union. Critical points partition the number line into regions whose signs can be tested.
Evidence boundary
A sign chart establishes interval behavior; it does not authorize cancellation across a zero denominator.
Common trap
Two boundary values imply every value between them works. The operator and sign pattern determine whether the middle, outside, or separate regions satisfy the condition.
Representations
inequality · number line · table
A3Exponential notation1 objective
Objective 1QR-MAT-EXP-01

Control exponential notation

Apply exponent and scientific-notation rules while preserving sign, base, reciprocal, and order-of-magnitude meaning.

Must know
Exponent rules apply to matching bases and to complete products or quotients, not arbitrary sums. A negative exponent creates a reciprocal; it does not make the value negative.
Evidence boundary
Keep exact form until the requested precision is known; do not manufacture significant digits.
Common trap
A negative exponent means a negative result. It indicates reciprocal power; the base determines the sign.
Representations
equation · table
A4Absolute value1 objective
Objective 1QR-MAT-ABS-01

Read absolute value as distance

Interpret absolute value as distance and solve equations or comparisons by separating the required cases.

Must know
Absolute value is a nonnegative distance from zero or another reference point. An equation |u| = c with c > 0 produces two cases; a negative c produces no real solution.
Evidence boundary
Case splitting must preserve the entire expression inside the absolute-value bars.
Common trap
Absolute value means delete a minus sign. It measures distance and may require two algebraic cases.
Representations
equation · number line · comparison
A5Ratios and proportions2 objectives
Objective 1QR-MAT-RAT-01

Normalize ratios and rates

Compare ratios, unit rates, percentages, and proportional quantities only after aligning the same base and units.

Must know
A ratio compares ordered quantities; swapping numerator and denominator changes its meaning. Percent change uses the original quantity as the denominator unless the prompt defines another base.
Evidence boundary
A similar-looking fraction is not comparable until both quantities and units refer to the same basis.
Common trap
Percent change is final minus initial divided by final. The ordinary comparison base is the original value.
Representations
equation · table · unit ledger
Objective 2QR-MAT-RAT-02

Scale proportional models

Solve direct, inverse, and multistep proportion problems and verify that the direction of change matches the model.

Must know
Direct proportionality keeps a ratio constant; inverse proportionality keeps a product constant. A reasonableness check should confirm whether increasing one quantity should increase or decrease the other.
Evidence boundary
Do not assume proportionality merely because two quantities change together.
Common trap
Every rate relationship scales directly. Some models are inverse or nonproportional; the stated relationship controls the equation.
Representations
equation · graph · unit ledger

ANSWER INTEGRITY

The key is recomputed.
The display may shuffle.

  1. 01Exact model

    Rational arithmetic stays exact before any display formatting.

  2. 02Semantic options

    Each choice carries a mathematical value separate from its visible wording.

  3. 03Unique match

    The computed result must match exactly one option or the build fails.

  4. 04Stable identity

    Question and answer order can change without changing correctness.

Free source ladder

Trace scope and instruction.

ADA DAT 2026 Candidate GuideSection timing, count, calculator, and administration policy. ↗ADA DAT User’s ManualMathematical-problem split and displayed Algebra topic lines. ↗OpenStax Algebra and Trigonometry 2eFree instruction for expressions, equations, inequalities, powers, and absolute value. ↗OpenStax Prealgebra 2eFree instruction for ratios, rates, proportions, and percentages. ↗