ACT Math Algebra: Building Skills in Equations, Functions, and Models

Linear Equations and Inequalities

What “linear” means (and why it matters)

A linear equation is an equation whose graph is a straight line. “Linear” basically means you never see variables multiplied together (like xy), variables in exponents (like 2^x), or variables under radicals (like \sqrt{x}). The power of linear equations is that they model constant-rate change—every time x increases by 1, y changes by the same amount.

A linear inequality is like a linear equation, but instead of asking for the exact point(s) where two expressions are equal, you’re describing a whole region of values where one expression is greater than or less than another.

On the ACT, linear equations and inequalities show up constantly because they’re the foundation for function thinking, graph interpretation, and setting up word problems.

Forms of a line and how to move between them

The most common forms you’ll see are:

  • Slope-intercept form (easy for graphing):

y = mx + b

Here m is the **slope** (rate of change) and b is the y-intercept (where the line crosses the y-axis).

  • Standard form (common in systems/word problems):

Ax + By = C

Usually A, B, and C are integers.

  • Point-slope form (useful when you know a point and a slope):

y - y_1 = m(x - x_1)

To avoid mistakes, keep the roles straight: x1 and y1 are the coordinates of the known point.

Slope as “rise over run”

Slope measures how steep a line is:

m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}

A frequent error is swapping points in the numerator but not the denominator (or vice versa). If you do swap, you must swap in both places so the signs stay consistent.

Solving linear equations: balancing, not guessing

Solving a linear equation means finding the value(s) of the variable that make the statement true. The key idea is equivalence: you can do the same operation to both sides without changing the solution.

Steps that keep you out of trouble:

  1. Distribute to remove parentheses.
  2. Combine like terms on each side.
  3. Move variable terms to one side.
  4. Move constants to the other.
  5. Divide by the coefficient of the variable.

Example 1 (multi-step equation):
Solve:

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

Distribute:

6x - 15 = 4x + 7

Move 4x to the left (subtract 4x from both sides):

2x - 15 = 7

Add 15 to both sides:

2x = 22

Divide by 2:

x = 11

A common mistake here is distributing incorrectly, especially with negatives (for instance turning 3(2x - 5) into 6x - 5).

Linear inequalities (including “flip the sign”)

Solving linear inequalities is almost identical to solving equations—except for one crucial rule:

If you multiply or divide both sides by a negative number, you must reverse the inequality sign.

Example 2 (inequality with a negative coefficient):
Solve:

-2x + 3 \le 11

Subtract 3 from both sides:

-2x \le 8

Divide by -2 and flip the sign:

x \ge -4

If you forget to flip the sign, you get the opposite set of solutions.

Compound inequalities and interval thinking

Sometimes you’ll see:

2 < 3x + 1 \le 10

You solve it like a “chain,” performing the same operations to all three parts.

Subtract 1:

1 < 3x \le 9

Divide by 3:

\frac{1}{3} < x \le 3

This means x is bigger than \frac{1}{3} and up to (including) 3.

Exam Focus
  • Typical question patterns:
    • Solve a linear equation or inequality with distribution and variables on both sides.
    • Interpret or compute slope/intercepts from an equation or two points.
    • Translate a short word problem into y = mx + b (rate and starting value).
  • Common mistakes:
    • Forgetting to flip the inequality when dividing/multiplying by a negative.
    • Distributing incorrectly (especially through subtraction).
    • Mixing up slope and intercept, or using inconsistent point order in the slope formula.

Systems of Equations

What a system is and what solutions mean

A system of equations is a set of equations that must be true at the same time. For two linear equations in two variables, a solution is an ordered pair (x, y) that makes both equations true.

Graphically, each equation is a line, and the solution is where the lines intersect:

  • One intersection point: one solution.
  • Parallel distinct lines: no solution.
  • Same line (equivalent equations): infinitely many solutions.

This matters because many real situations involve two constraints—like cost and quantity, or distance and time for two moving objects.

Solving by substitution

Substitution is best when one equation is already solved for one variable (or can be easily).

Example 1 (substitution):

y = 2x + 1

3x + y = 16

Substitute 2x + 1 for y in the second equation:

3x + (2x + 1) = 16

Combine like terms:

5x + 1 = 16

Subtract 1:

5x = 15

Divide by 5:

x = 3

Now plug back into y = 2x + 1:

y = 2(3) + 1 = 7

Solution: (3, 7) .

A common mistake is plugging back into the wrong equation after algebra errors; always use the equation that’s simplest to check your work.

Solving by elimination (addition method)

Elimination is best when you can line up coefficients to cancel a variable by adding or subtracting equations.

Example 2 (elimination):

2x + 3y = 13

2x - y = 5

Subtract the second equation from the first to eliminate x:

(2x + 3y) - (2x - y) = 13 - 5

Simplify:

4y = 8

So:

y = 2

Plug into 2x - y = 5:

2x - 2 = 5

2x = 7

x = \frac{7}{2}

Solution: \left(\frac{7}{2}, 2\right) .

Common pitfall: when subtracting equations, failing to distribute the negative across all terms in parentheses.

Systems in word problems

Systems often come from “two conditions” problems:

  • Tickets: adult/child tickets with totals.
  • Mixtures: two solutions with different concentrations.
  • Motion: two speeds and a time/distance relationship.

A reliable setup pattern:

  1. Define variables clearly (what does x represent? what does y represent?).
  2. Write one equation per condition.
  3. Solve and interpret (check if the values make sense in context).
Exam Focus
  • Typical question patterns:
    • Solve a system by substitution or elimination.
    • Determine the number of solutions from slopes/graph descriptions.
    • Set up a system from a word problem (totals and rates are common).
  • Common mistakes:
    • Algebra sign errors when combining equations (especially subtracting).
    • Solving for one variable but forgetting to substitute back to get the other.
    • Interpreting x and y incorrectly in word problems (units mismatch).

Polynomial Expressions and Equations

What polynomials are

A polynomial is an expression made by adding/subtracting terms of the form ax^n where a is a constant and n is a nonnegative integer. Examples include:

4x^3 - 2x + 7

Polynomials matter because they’re the language of many algebra problems: factoring, solving equations, modeling area/volume, and analyzing graphs all depend on polynomial structure.

Key vocabulary:

  • Term: a piece like -2x or 7.
  • Coefficient: the number multiplying the variable part (in -2x, coefficient is -2).
  • Degree: the highest exponent (degree 3 in 4x^3 - 2x + 7).

Operations with polynomials: structure over memorization

Combining like terms

You can only combine terms with exactly the same variable part.

Example:

3x^2 + 5x - 2 + 4x^2 - x + 9 = 7x^2 + 4x + 7

A common mistake is combining x and x^2 terms, which are not like terms.

Distributive property and multiplication

Multiplying polynomials relies on distributing every term.

Example 1 (binomial times binomial):
Expand:

(x + 3)(x - 5)

Distribute:

x(x - 5) + 3(x - 5)

x^2 - 5x + 3x - 15

Combine like terms:

x^2 - 2x - 15

Students often drop a term or miss a negative during distribution.

Polynomial equations and factoring as the main solving tool

A polynomial equation sets a polynomial equal to zero or to another expression. Often, the fastest path is to rearrange into:

\text{polynomial} = 0

Then factor and use the zero product principle:

If ab = 0, then a = 0 or b = 0.

Example 2 (solve by factoring):

x^2 - 2x - 15 = 0

Factor:

(x - 5)(x + 3) = 0

So:

x - 5 = 0 \Rightarrow x = 5

x + 3 = 0 \Rightarrow x = -3

Exam Focus
  • Typical question patterns:
    • Simplify polynomial expressions (combine like terms, distribute).
    • Expand products like (x + a)(x + b).
    • Solve a polynomial equation by setting it equal to zero and factoring.
  • Common mistakes:
    • Combining unlike terms (such as x with x^2).
    • Sign mistakes when distributing negatives.
    • Forgetting to set the equation to zero before using factoring/zero product.

Quadratic Equations and Factoring

What quadratics are and how they behave

A quadratic is a polynomial of degree 2:

ax^2 + bx + c

with a \ne 0. Quadratics matter because they model curved relationships—projectile motion, area problems, profit models—and they appear frequently in algebra questions that require solving, factoring, or interpreting graphs.

Graphically, quadratics are parabolas. Key features:

  • If a > 0, the parabola opens up (has a minimum).
  • If a < 0, it opens down (has a maximum).

Factoring as “reverse distributing”

Factoring turns a quadratic into a product of simpler expressions, which then lets you solve using the zero product principle.

Factoring x^2 + bx + c

You look for two numbers that multiply to c and add to b.

Example 1:
Factor:

x^2 + 7x + 12

Find numbers that multiply to 12 and add to 7: 3 and 4.

(x + 3)(x + 4)

A common mistake is choosing numbers that multiply correctly but add to the wrong sign.

Factoring ax^2 + bx + c where a \ne 1

One reliable method is grouping after finding factors of ac that sum to b.

Example 2:
Factor:

2x^2 + 7x + 3

Compute ac = 2 \cdot 3 = 6. Find numbers that multiply to 6 and add to 7: 6 and 1.

Split the middle term:

2x^2 + 6x + x + 3

Group:

(2x^2 + 6x) + (x + 3)

Factor each group:

2x(x + 3) + 1(x + 3)

Factor out the common binomial:

(2x + 1)(x + 3)

Solving quadratics: factoring, square roots, or formula

Factoring is fastest when it works nicely, but not every quadratic factors over integers.

Solving by taking square roots (when there’s no middle term)

If you can isolate x^2, you can take square roots.

Example:

x^2 = 20

x = \pm \sqrt{20} = \pm 2\sqrt{5}

Common mistake: forgetting the \pm.

Quadratic formula

For:

ax^2 + bx + c = 0

the solutions are:

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

The expression under the square root is the discriminant:

D = b^2 - 4ac

  • If D > 0: two real solutions.
  • If D = 0: one real solution (a repeated root).
  • If D < 0: no real solutions (not typically emphasized unless asked).

A common ACT error is substituting a, b, c with wrong signs—always read them from the equation written as ax^2 + bx + c = 0.

Exam Focus
  • Typical question patterns:
    • Factor a quadratic and then solve for zeros.
    • Recognize a quadratic in disguised form (expand first, then factor).
    • Use the quadratic formula when factoring is hard.
  • Common mistakes:
    • Dropping the \pm when taking square roots.
    • Using incorrect signs for b or c in the quadratic formula.
    • Factoring errors from guessing factor pairs without checking the middle term.

Radical Expressions and Equations

What radicals represent

A radical expression involves roots like square roots. The square root of a number is a value that, when squared, gives the original number.

For nonnegative real numbers:

\sqrt{a^2} = |a|

That absolute value is important: the principal square root is nonnegative, so \sqrt{(-3)^2} = 3, not -3.

Radicals matter on the ACT because they test your comfort with equivalence transformations, simplifying expressions, and solving equations that involve squaring.

Simplifying radicals: pulling out perfect squares

To simplify \sqrt{n}, factor n into a perfect square times something else.

Example 1:

\sqrt{72}

Factor:

72 = 36 \cdot 2

So:

\sqrt{72} = \sqrt{36}\sqrt{2} = 6\sqrt{2}

A common mistake is trying to split a sum under a radical (you cannot do this):

\sqrt{a + b} \ne \sqrt{a} + \sqrt{b}

Rational exponents connection

Radicals and exponents are two notations for the same idea:

\sqrt{x} = x^{\frac{1}{2}}

\sqrt[n]{x} = x^{\frac{1}{n}}

This is useful when simplifying expressions like x^{3/2}.

Solving radical equations (and the need to check)

To solve an equation with radicals, you often isolate the radical and square both sides. Squaring is powerful but dangerous because it can introduce extraneous solutions (answers that satisfy the squared equation but not the original).

Example 2 (radical equation with an extraneous check):
Solve:

\sqrt{x + 5} = x - 1

First, note a domain restriction: right side must be nonnegative.

x - 1 \ge 0 \Rightarrow x \ge 1

Square both sides:

x + 5 = (x - 1)^2

Expand:

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

Bring all terms to one side:

0 = x^2 - 3x - 4

Factor:

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

So x = 4 or x = -1. Now check in the original equation:

  • For x = 4:

\sqrt{9} = 3 and 4 - 1 = 3, works.

  • For x = -1:

\sqrt{4} = 2 but -1 - 1 = -2, does not work.

So the solution is:

x = 4

Exam Focus
  • Typical question patterns:
    • Simplify radicals by factoring out perfect squares.
    • Solve a basic radical equation by squaring.
    • Rewrite between radical and rational exponent forms.
  • Common mistakes:
    • Failing to check for extraneous solutions after squaring.
    • Incorrectly distributing a square: (a - b)^2 is not a^2 - b^2.
    • Assuming \sqrt{a + b} splits into \sqrt{a} + \sqrt{b}.

Exponential Equations and Growth/Decay

What makes something exponential

An exponential expression has a variable in the exponent, like 2^x. Exponential models matter because they represent multiplicative change: growing by a fixed percent, doubling, halving, repeated compounding.

Contrast this with linear change:

  • Linear: add the same amount each step.
  • Exponential: multiply by the same factor each step.

Exponent rules (the engine behind simplification)

These rules help you rewrite expressions into solvable forms:

a^m a^n = a^{m+n}

\frac{a^m}{a^n} = a^{m-n}

(a^m)^n = a^{mn}

a^0 = 1

(assuming a \ne 0)

A common mistake is thinking (a + b)^n = a^n + b^n , which is not true in general.

Solving basic exponential equations

On the ACT, many exponential equations are designed so you can rewrite both sides with the same base.

Example 1:
Solve:

2^{x+1} = 16

Rewrite 16 as a power of 2:

16 = 2^4

So:

2^{x+1} = 2^4

Match exponents:

x + 1 = 4

x = 3

If bases cannot be matched nicely, ACT problems may still avoid requiring logarithms by providing answer choices that you can test or by creating structures that simplify.

Growth and decay models

A common form for exponential growth/decay is:

A = A_0 (1 + r)^t

  • A_0 is the initial amount.
  • r is the growth rate per period (as a decimal). If it’s decay, r is negative or you use 1 - r with r positive as a decay rate.
  • t is the number of periods.

Example 2 (decay):
A value decreases by 20% each year. If it starts at 500, what is it after 3 years?

Decay factor per year:

1 - 0.20 = 0.80

Compute:

A = 500(0.8)^3

A = 500(0.512) = 256

A typical mistake is subtracting 20% of the original each year (linear thinking) instead of multiplying by 0.8 each year (exponential thinking).

Exam Focus
  • Typical question patterns:
    • Solve exponential equations by rewriting with common bases.
    • Interpret growth/decay in words and translate to A = A_0(1 + r)^t.
    • Compare linear versus exponential models from tables or scenarios.
  • Common mistakes:
    • Treating percent change as additive instead of multiplicative.
    • Misreading r: using 20 instead of 0.20.
    • Applying exponent rules incorrectly, especially with sums in parentheses.

Graphing and Modeling Expressions

Why graphs are central in algebra

A graph is a picture of a relationship. On ACT Math, graphs are not just about plotting points—they’re about interpreting meaning: rate, intercepts, turning points, and where two expressions are equal.

A model is a mathematical expression used to represent a real situation. Modeling questions test whether you can translate between words, tables, equations, and graphs.

Function notation and evaluating expressions

A function assigns each input exactly one output. If you see:

f(x) = 2x - 3

then f(5) means “plug in 5 for x”:

f(5) = 2(5) - 3 = 7

A common mistake is reading f(x) as f \cdot x; it is not multiplication.

Graphing lines: intercepts and slope as meaning

To graph:

y = mx + b

  • Plot b on the y-axis.
  • Use slope m as “rise/run” to get a second point.
  • Draw the line through points.

If the equation is in standard form, intercepts can be fast:

  • x-intercept: set y = 0.
  • y-intercept: set x = 0.

Example 1 (intercepts):
Find intercepts of:

3x + 2y = 12

Set y = 0:

3x = 12 \Rightarrow x = 4

So x-intercept is (4, 0) .

Set x = 0:

2y = 12 \Rightarrow y = 6

So y-intercept is (0, 6) .

Graphing quadratics: vertex and intercepts

For:

y = ax^2 + bx + c

  • The y-intercept is c (point (0, c) ).
  • The x-intercepts are solutions to ax^2 + bx + c = 0 (if real).

The vertex occurs at:

x = \frac{-b}{2a}

Then you plug that x back into the equation to get the vertex y-value.

Example 2 (vertex):
For:

y = x^2 - 6x + 5

Compute vertex x-coordinate:

x = \frac{-(-6)}{2(1)} = 3

Plug in x = 3:

y = 3^2 - 6(3) + 5 = 9 - 18 + 5 = -4

Vertex is (3, -4) .

A common mistake is using x = \frac{-b}{a} (missing the 2).

Modeling from words: identifying the type

Many ACT modeling problems are really “identify the relationship” problems:

  • Linear if you hear “constant increase/decrease,” “per,” “each,” “fixed rate.”
  • Exponential if you hear “percent,” “multiplies,” “doubles every,” “compound.”
  • Quadratic if you see area with one dimension depending on another, or a symmetric “maximum/minimum” context.

When building a model, pay attention to units:

  • If m is dollars per hour, then mx is dollars when x is hours.
  • If r is percent per year, t must be years in A = A_0(1 + r)^t.

Intersections as solutions

When two graphs intersect, the intersection points represent solutions to the system:

f(x) = g(x)

Graph interpretation often replaces algebraic solving: you might be asked for an x-value where two curves meet or where a line crosses an axis.

Exam Focus
  • Typical question patterns:
    • Interpret slope/intercepts or read them off a graph.
    • Find where a graph crosses the axes or where two graphs intersect.
    • Choose an equation that matches a given graph or scenario (linear vs quadratic vs exponential).
  • Common mistakes:
    • Confusing x-intercepts (where y = 0) with y-intercepts (where x = 0).
    • Misidentifying exponential versus linear change in word problems.
    • Evaluating function notation incorrectly (treating f(x) like multiplication).