Model Comparison: Algebra

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Gemini 3 Pro

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GPT 5.2 Pro

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What You Need to Know

• Algebra on ACT Math is mostly about solving (equations/inequalities), interpreting (what a solution means), and modeling (turning words/data into expressions, equations, or graphs).

• Master core manipulations: distributing, combining like terms, factoring, and keeping operations balanced—most mistakes come from small algebra slips.

• Expect multi-step word problems that hide algebra inside contexts (rates, mixtures, money, geometry formulas) and ask for the “best” choice.

Linear Equations and Inequalities

Core ideas

A linear equation is an equation where variables have exponent 1 (no products of variables). In slope-intercept form:

y = mx + b

• m = slope (rate of change)

• b = y-intercept (value when x = 0)

A linear inequality uses <, \le, >, \ge and its solution is a range of values.

Key skills

• Solving one-variable equations by inverse operations (keep both sides balanced).

• Solving inequalities—when multiplying/dividing by a negative, flip the inequality.

• Translating words:

  • “at least” \rightarrow \ge

  • “no more than” \rightarrow \le

  • “between” often means a compound inequality like a < x < b

Worked examples

1) Linear equation

Solve 3(2x - 5) = 4x + 7

6x - 15 = 4x + 7

2x - 15 = 7

2x = 22

x = 11

2) Inequality (flip sign)

Solve -2x + 3 \ge 11

-2x \ge 8

Divide by -2 (flip):

x \le -4

Exam Focus

• Why it matters: Linear solving is one of the most frequent ACT algebra tasks and appears in both pure algebra and word problems.

• Typical question patterns:

  • Solve for x from a multi-step equation with parentheses/fractions.

  • Interpret slope/rate from a context (e.g., dollars per hour).

  • Solve/graph an inequality or choose a value that satisfies it.

• Common mistakes:

  • Forgetting to flip the inequality when dividing by a negative.

  • Distributing incorrectly: a(b + c) \ne ab + c.

  • Dropping a negative when moving terms across the equals sign (use addition/subtraction to move terms safely).

Systems of Equations

Core ideas

A system of equations is two (or more) equations with the same variables; solutions are values that satisfy all equations.

Common methods:

• Substitution: solve one equation for a variable, plug into the other.

• Elimination: add/subtract equations to cancel a variable.

Interpretation (graphically): intersection point(s) of lines.

• One solution: lines intersect once.

• No solution: parallel distinct lines.

• Infinitely many: same line.

Worked example (elimination)

Solve:

\begin{cases}
2x + y = 11\
3x - y = 4
\end{cases}

Add equations to eliminate y:

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

5x = 15

x = 3

Substitute into 2x + y = 11:

2(3) + y = 11

y = 5

Solution: (3,5)

ACT-style modeling note

Many system problems are word problems: e.g., ticket sales, mixtures, or comparing two plans. Define variables clearly (e.g., a = adult tickets, c = child tickets).

Exam Focus

• Why it matters: Systems test algebraic setup from words and clean execution—high-value because answers are multiple-choice and errors are diagnosable.

• Typical question patterns:

  • Solve a system and select the ordered pair.

  • Determine whether a system has 0, 1, or infinitely many solutions.

  • Build a system from a context (cost/rate/mixture).

• Common mistakes:

  • Sign errors when adding equations in elimination.

  • Solving for one variable incorrectly before substitution.

  • Forgetting that the final answer may ask for one variable (e.g., “how many adults?”) not the ordered pair.

Polynomial Expressions and Equations

Core ideas

A polynomial is a sum of terms a_n x^n with nonnegative integer exponents (e.g., x^3, x^2).

Key operations:

• Combine like terms.

• Multiply polynomials (distribute / FOIL for binomials).

• Factor polynomials to solve equations.

Important identities

a^2 - b^2 = (a-b)(a+b)

(a+b)^2 = a^2 + 2ab + b^2

(a-b)^2 = a^2 - 2ab + b^2

Worked example (factoring to solve)

Solve x^3 - 4x = 0

Factor out the GCF:

x(x^2 - 4) = 0

Difference of squares:

x(x-2)(x+2) = 0

So x = 0 or x = 2 or x = -2.

Exam Focus

• Why it matters: Polynomial manipulation underlies factoring, solving, simplifying rational expressions, and function behavior questions.

• Typical question patterns:

  • Expand and simplify an expression, then match an equivalent form.

  • Factor completely (often GCF first, then special products).

  • Solve polynomial equations by factoring and using the zero-product rule.

• Common mistakes:

  • Skipping the GCF and making factoring harder.

  • Incorrectly applying identities (especially signs in a^2 - b^2).

  • Forgetting that solving ab = 0 means a = 0 or b = 0.

Quadratic Equations and Factoring

Core ideas

A quadratic has the form:

ax^2 + bx + c = 0

Ways to solve:

• Factoring (fastest when it works)

• Quadratic formula (works always)

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

The discriminant D = b^2 - 4ac tells the number of real solutions:

• D > 0: two real solutions

• D = 0: one real solution

• D < 0: no real solutions

Worked examples

1) Factoring

Solve x^2 - 5x + 6 = 0

(x-2)(x-3) = 0

x = 2 or x = 3

2) Quadratic formula

Solve 2x^2 + 3x - 2 = 0

x = \frac{-3 \pm \sqrt{3^2 - 4(2)(-2)}}{2(2)}

x = \frac{-3 \pm \sqrt{9 + 16}}{4}

x = \frac{-3 \pm 5}{4}

x = \frac{1}{2} or x = -2

Exam Focus

• Why it matters: Quadratics appear in function/graph questions, optimization-style word problems, and algebraic solving.

• Typical question patterns:

  • Solve a quadratic by factoring or the quadratic formula.

  • Use the discriminant to decide how many real solutions.

  • Match equivalent forms (standard vs factored) to read roots.

• Common mistakes:

  • Sign errors in factoring (especially when c < 0).

  • Incorrect substitution into the quadratic formula (mixing up a, b, c).

  • Forgetting the \pm and giving only one solution.

Radical Expressions and Equations

Core ideas

A radical involves roots like \sqrt{x}. Key property (for nonnegative values):

\sqrt{ab} = \sqrt{a}\sqrt{b}

Simplify radicals by factoring out perfect squares.

Rationalizing denominators

If you have \frac{1}{\sqrt{a}}, multiply top and bottom by \sqrt{a}:

\frac{1}{\sqrt{a}} \cdot \frac{\sqrt{a}}{\sqrt{a}} = \frac{\sqrt{a}}{a}

Solving radical equations (watch for extraneous solutions)

Typical steps:
1) Isolate the radical.
2) Square both sides.
3) Solve the resulting equation.
4) Check in the original equation.

Worked example

Solve \sqrt{x + 5} = x - 1

Domain: need x - 1 \ge 0 so x \ge 1.

Square both sides:

x + 5 = (x - 1)^2

x + 5 = x^2 - 2x + 1

0 = x^2 - 3x - 4

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

Candidates: x = 4 or x = -1.

Check domain and original: x = -1 fails domain. x = 4 works.

Exam Focus

• Why it matters: Radicals test algebraic manipulation plus reasoning about domains—ACT often includes “which is equivalent?” simplification items.

• Typical question patterns:

  • Simplify a radical expression.

  • Rationalize a denominator.

  • Solve an equation with a square root and choose the valid solution.

• Common mistakes:

  • Not checking for extraneous solutions after squaring.

  • Treating \sqrt{a+b} as \sqrt{a} + \sqrt{b} (not generally true).

  • Forgetting domain restrictions (like requiring the radicand to be nonnegative).

Exponential Equations and Growth/Decay

Core ideas

An exponential function has the variable in the exponent, like:

y = a\,b^x

• If b > 1: growth

• If 0 < b < 1: decay

Common models:

• Percent growth per step: y = a(1+r)^t

• Percent decay per step: y = a(1-r)^t

Solving exponential equations (ACT level)

ACT problems often use rewrite-to-same-base strategies.

Example: Solve 2^{x+1} = 16

Rewrite 16 = 2^4:

2^{x+1} = 2^4

So x + 1 = 4, hence x = 3.

Worked word example (growth)

A population is 5000 and grows by 3\% per year. After t years:

P(t) = 5000(1.03)^t

Exam Focus

• Why it matters: Exponential growth/decay connects algebra to real contexts (interest, depreciation, populations) and tests your ability to interpret parameters.

• Typical question patterns:

  • Match a word description to an exponential expression.

  • Solve simple exponential equations by rewriting bases.

  • Compare growth rates (which increases faster?).

• Common mistakes:

  • Using 1 + r vs 1 - r incorrectly (growth vs decay).

  • Confusing the initial value a with the rate/base.

  • Treating exponential change as linear (adding a constant instead of multiplying by a factor).

Graphing and Modeling Expressions

Core ideas

Modeling means representing a relationship with an expression/equation and using it to answer questions.

Graphing on ACT is mostly interpretive (read slopes/intercepts, compare functions, identify transformations).

Linear graph features

From y = mx + b:

• m: rise/run (rate)

• b: y-intercept

Given two points (x1, y1) and (x2, y2):

m = \frac{y2 - y1}{x2 - x1}

Point-slope form:

y - y1 = m(x - x1)

Quadratic graph features

In vertex form:

y = a(x - h)^2 + k

• Vertex at (h, k)

• a > 0 opens up; a < 0 opens down

Transformations (quick)

If f(x) is a base function:

• f(x) + k shifts up by k

• f(x - h) shifts right by h

• -f(x) reflects across the x-axis

Worked example (model from words)

A taxi charges a base fee of 3 dollars plus 2 dollars per mile. Let m be miles and C be cost:

C = 2m + 3

If m = 7:

C = 2(7) + 3 = 17

Exam Focus

• Why it matters: ACT often gives a graph/table and asks you to interpret, compare, or select the matching equation—fast points if you recognize features.

• Typical question patterns:

  • Identify slope/intercept from a graph or context.

  • Choose which equation matches a graph (or which graph matches an equation).

  • Interpret function changes (shift, reflect, stretch) from an equation.

• Common mistakes:

  • Mixing up slope and intercept when reading graphs.

  • Shifts: confusing f(x - h) (right) with f(x) - h (down).

  • Using two points but swapping differences inconsistently (keep numerator and denominator aligned).

Quick Review Checklist

• Can you solve multi-step linear equations and simplify expressions with distribution?

• Can you solve linear inequalities and remember to flip the sign when dividing by a negative?

• Can you solve a system using substitution or elimination and interpret what the solution means?

• Can you factor polynomials using GCF, difference of squares, and trinomial factoring?

• Can you solve quadratic equations by factoring and by using x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}?

• Can you simplify radicals, rationalize denominators, and check for extraneous solutions?

• Do you know how to set up exponential growth/decay models like a(1\pm r)^t?

• Can you read slope/intercept from graphs and match equations to graphs using key features?

Final Exam Pitfalls

1. Forgetting restrictions (radicals/denominators): always check domains (e.g., radicand \ge 0, denominator \ne 0) and verify solutions after squaring.

2. Inequality sign errors: when multiplying/dividing by a negative, flip < to > (and \le to \ge).

3. Factoring shortcuts without checking: factor out the GCF first, then apply patterns like a^2 - b^2—don’t skip steps.

4. Quadratic formula mislabeling: identify a, b, c from ax^2 + bx + c before substituting; keep parentheses in -b \pm \sqrt{b^2 - 4ac}.

5. Modeling mismatch: in word problems, confirm units and meaning—your equation should reflect “per” rates (slope) and fixed starting amounts (intercept).

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Claude Opus 4.6

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What You Need to Know

• Algebra accounts for roughly 12–15 questions (about 20–25%) of the 60-question ACT Math test. Mastering this section is one of the highest-impact things you can do for your score.

• The ACT tests your ability to manipulate expressions, solve equations, and interpret graphs — almost always in a multiple-choice format where plugging in answer choices or back-solving is a viable strategy.

• You must be comfortable moving between algebraic, graphical, and verbal representations of the same relationship — many questions ask you to translate a word problem into an equation or identify a graph.

• Speed matters: you have roughly 1 minute per question, so fluency with core algebraic techniques (distributing, factoring, combining like terms) is essential.

Linear Equations and Inequalities

A linear equation is any equation that can be written in the form ax + b = c, where a, b, and c are constants. The graph of a linear equation in two variables is a straight line.

Key Forms

Inequalities

• Solve exactly like equations except: when you multiply or divide by a negative number, flip the inequality sign.

• Example: Solve -2x + 5 > 11

  1. Subtract 5: -2x > 6

  2. Divide by -2 and flip: x < -3

Exam Focus

• Why it matters: Linear equations appear on virtually every ACT. They're the foundation of the algebra section.

• Typical question patterns:

  • "What is the slope of the line passing through (2, 5) and (6, 13)?" — use m = \frac{y2 - y1}{x2 - x1}

  • Solving a multi-step equation for x with fractions or decimals

  • Interpreting slope and intercept in a real-world context (e.g., "the monthly fee" = y-intercept, "cost per item" = slope)

• Common mistakes:

  • Forgetting to flip the inequality sign when dividing by a negative

  • Confusing slope with y-intercept when reading y = mx + b

  • Sign errors when distributing negatives across parentheses

Systems of Equations

A system of equations is a set of two or more equations with the same variables. On the ACT, you'll almost always see systems of two linear equations in two unknowns.

Solving Methods

1. Substitution: Solve one equation for a variable, then substitute into the other.

2. Elimination (addition/subtraction): Multiply equations so that one variable cancels when you add the equations together.

Example — Solve:
2x + 3y = 12
x - y = 1

Using substitution: from the second equation, x = y + 1. Substitute into the first:
2(y + 1) + 3y = 12 \implies 2y + 2 + 3y = 12 \implies 5y = 10 \implies y = 2, so x = 3.

Special Cases

• No solution: The lines are parallel (same slope, different intercepts).

• Infinitely many solutions: The equations represent the same line.

Exam Focus

• Why it matters: 2–3 questions per test typically involve systems.

• Typical question patterns:

  • Word problems that set up two relationships (e.g., ticket prices, mixture problems)

  • Asking for a specific expression like x + y rather than individual values — look for shortcuts

  • Questions about how many solutions exist

• Common mistakes:

  • Solving for x when the question asks for y (or vice versa)

  • Arithmetic errors in elimination — double-check your multiplied equations

  • Not recognizing when a shortcut exists (e.g., the question asks for 2x + 3y directly)

Polynomial Expressions and Equations

Polynomials are expressions with one or more terms of the form ax^n, where n is a non-negative integer. You need to be fluent with:

• Adding/subtracting: Combine like terms

• Multiplying: Use the distributive property (FOIL for binomials)

• Dividing: Polynomial long division or synthetic division (rare on the ACT)

FOIL example: (x + 3)(x - 5) = x^2 - 5x + 3x - 15 = x^2 - 2x - 15

Key Identities

• a^2 - b^2 = (a + b)(a - b) — difference of squares

• (a + b)^2 = a^2 + 2ab + b^2

• (a - b)^2 = a^2 - 2ab + b^2

Exam Focus

• Why it matters: Simplifying and manipulating polynomials underpins factoring and quadratics.

• Typical question patterns:

  • Simplify a product of binomials

  • Factor using difference of squares

  • Find the value of a polynomial expression for a given x

• Common mistakes:

  • Forgetting the middle term: (x + 3)^2 \neq x^2 + 9; it's x^2 + 6x + 9

  • Sign errors when distributing a negative across parentheses

Quadratic Equations and Factoring

A quadratic equation has the standard form ax^2 + bx + c = 0. There are three primary solving methods:

1. Factoring: Express as (x - r)(x - s) = 0, then x = r or x = s.

2. Quadratic formula: x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

3. Completing the square: Rewrite in vertex form (x - h)^2 = k.

The discriminant D = b^2 - 4ac tells you:

Memory aid: "Discriminant Determines Destiny" — positive means two, zero means one, negative means none.

Exam Focus

• Why it matters: Quadratics are among the most frequently tested algebra topics on the ACT (2–4 questions).

• Typical question patterns:

  • "What are the solutions to x^2 - 5x + 6 = 0?" — factor to (x-2)(x-3) = 0

  • Identifying the vertex or axis of symmetry: x = \frac{-b}{2a}

  • Word problems involving area or projectile motion

• Common mistakes:

  • Forgetting to set the equation equal to zero before factoring

  • Sign errors in the quadratic formula — especially with -b

  • Confusing roots/solutions/zeros/x-intercepts — they all mean the same thing

Radical Expressions and Equations

A radical expression contains a root, most commonly a square root \sqrt{x}.

Solving Radical Equations

1. Isolate the radical on one side.

2. Square both sides (or raise to the appropriate power).

3. Solve the resulting equation.

4. Check for extraneous solutions — squaring can introduce false answers.

Example: Solve \sqrt{x + 3} = 5

1. Square both sides: x + 3 = 25

2. Solve: x = 22

3. Check: \sqrt{22 + 3} = \sqrt{25} = 5 ✓

Simplifying Radicals

• \sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}

• \sqrt{a} \cdot \sqrt{b} = \sqrt{ab}

Exam Focus

• Why it matters: Usually 1–2 questions; they tend to be in the middle difficulty range.

• Typical question patterns:

  • Simplify a radical expression

  • Solve a radical equation and identify the valid solution

  • Rationalize a denominator: multiply by \frac{\sqrt{a}}{\sqrt{a}}

• Common mistakes:

  • Forgetting to check for extraneous solutions after squaring

  • Incorrectly simplifying: \sqrt{a + b} \neq \sqrt{a} + \sqrt{b}

  • Not fully simplifying the radical (e.g., leaving \sqrt{12} instead of 2\sqrt{3})

Exponential Equations and Growth/Decay

Exponential functions have the general form y = a \cdot b^x, where a is the initial value and b is the base (growth/decay factor).

• If b > 1: exponential growth

• If 0 < b < 1: exponential decay

Growth/Decay Models

A = A0(1 + r)^t for growth and A = A0(1 - r)^t for decay, where r is the rate and t is time.

Exponent Rules You Must Know

Exam Focus

• Why it matters: 1–2 questions, often framed as real-world scenarios (population, depreciation, compound interest).

• Typical question patterns:

  • "A population doubles every 5 years. If it starts at 1000, what is the population after 15 years?" — 1000 \cdot 2^3 = 8000

  • Simplify an expression using exponent rules

  • Identify whether a table of values represents linear or exponential growth

• Common mistakes:

  • Confusing linear growth (adding a constant) with exponential growth (multiplying by a constant)

  • Misapplying exponent rules: a^m \cdot b^m \neq (ab)^{2m}; it equals (ab)^m

  • Forgetting that the base in a decay problem is (1 - r), not r alone

Graphing and Modeling Expressions

This topic connects algebra to visual and contextual interpretation. You should be able to:

• Graph linear functions: Plot using slope and y-intercept or two points.

• Graph quadratics: Identify the vertex using x = \frac{-b}{2a}, direction of opening (a > 0 opens up, a < 0 opens down), and intercepts.

• Interpret graphs: Read off key features — intercepts, slope, vertex, increasing/decreasing intervals.

• Model real-world situations: Translate a word problem into an equation or choose the correct equation from answer choices.

Linear vs. Quadratic vs. Exponential — How to Tell

Exam Focus

• Why it matters: The ACT frequently tests whether you can connect an equation to its graph or to a real-world scenario — this appears across difficulty levels.

• Typical question patterns:

  • "Which equation best models the data in the table?"

  • "What does the y-intercept represent in this context?"

  • Matching a graph to an equation based on slope sign, intercepts, or curvature

• Common mistakes:

  • Mixing up which feature of the equation corresponds to which feature of the graph

  • Ignoring the context — the ACT often asks what a number means, not just what it equals

  • Not checking all answer choices against the graph's key features

Quick Review Checklist

• ☐ Can you solve a multi-step linear equation with fractions or parentheses?

• ☐ Do you know when to flip the inequality sign?

• ☐ Can you solve a system of equations using both substitution and elimination?

• ☐ Can you factor a quadratic expression, including difference of squares?

• ☐ Do you know the quadratic formula and what the discriminant tells you?

• ☐ Can you simplify radical expressions like \sqrt{72}?

• ☐ Do you remember to check for extraneous solutions in radical equations?

• ☐ Can you apply all the exponent rules fluently?

• ☐ Can you distinguish between linear, quadratic, and exponential growth from a table or graph?

• ☐ Can you translate a word problem into an algebraic equation?

Final Exam Pitfalls

1. Not reading what the question asks for. You solve for x perfectly, but the question asks for 2x + 1. Always re-read the question before selecting an answer.

2. Forgetting to flip the inequality sign. When multiplying or dividing both sides of an inequality by a negative number, the direction of the sign reverses. This is tested repeatedly.

3. Expanding (a + b)^2 incorrectly. It is a^2 + 2ab + b^2, not a^2 + b^2. The middle term is the most commonly forgotten piece.

4. Not setting a quadratic equal to zero before factoring. If you have x^2 - 5x = 6, you must rewrite as x^2 - 5x - 6 = 0 before factoring. Factoring both sides separately leads to wrong answers.

5. Accepting extraneous solutions in radical equations. Always plug your answer back into the original equation. Squaring both sides can create solutions that don't actually work.

6. Confusing growth rate with growth factor. A 15% growth rate means the growth factor is 1.15, not 0.15. The equation is A = A0(1.15)^t, not A = A0(0.15)^t.