ACT Math: Elementary Algebra

Elementary algebra forms the backbone of mathematical reasoning tested on the ACT, accounting for roughly 15–20% of the Math section. Mastering these concepts is non-negotiable for a high score, as they are the fundamental tools you will use to solve a wide array of problems, from simple equations to complex word problems.

Algebraic Expressions and Operations

An algebraic expression is a mathematical phrase that can contain numbers, variables (like $x$ or $y$), and operation symbols. The core skill here is evaluating these expressions, which means substituting a given number for the variable and then calculating the result using the correct order of operations (PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction).

For example, if $x=-2$, evaluate $3x^2-5x+1$.

1. Substitute: $3(-2{)}^2-5(-2)+1$
2. Exponents: $3(4)-5(-2)+1$
3. Multiplication: $12-(-10)+1$
4. Simplify: $12+10+1=23$

You must also be adept at simplifying expressions by combining like terms, which are terms that have the same variable raised to the same power. For instance, in $5x+3y-2x+4$, the like terms are $5x$ and $-2x$. Combining them gives $3x+3y+4$.

Solving Linear Equations and Inequalities

A linear equation is a statement that two algebraic expressions are equal. Your goal is to isolate the variable on one side of the equals sign. The golden rule is to perform the same operation to both sides. For multi-step equations, simplify each side first (distribute, combine like terms), then use inverse operations strategically.

Solve for $x$: $2(3x-4)=5x+6$.

1. Distribute: $6x-8=5x+6$
2. Subtract $5x$ from both sides: $x-8=6$
3. Add $8$ to both sides: $x=14$

A linear inequality is similar but uses symbols like $<$, $>$, $\le$, or $\ge$. You solve them just like equations, with one critical exception: if you multiply or divide both sides by a negative number, you must reverse the inequality sign. This is a classic trap on the ACT.

Solve for $y$: $-3y+7\ge 16$.

1. Subtract $7$: $-3y\ge 9$
2. Divide by $-3$ (and reverse the sign!): $y\le -3$

The solution set includes all numbers less than or equal to $-3$.

Working with Exponents and Polynomials

The rules of exponents are procedural and must be memorized. Key rules for the ACT include:

• Product Rule: $x^a\cdot x^b=x^{a+b}$
• Quotient Rule: $x^a/x^b=x^{a-b}$
• Power Rule: $(x^a{)}^b=x^{ab}$
• Negative Exponent: $x^{-a}=1/x^a$
• Zero Exponent: $x^0=1$ (where $x\ne 0$)

A polynomial is an expression with one or more terms involving variables with non-negative integer exponents. Core operations are addition, subtraction, and multiplication. For addition/subtraction, only combine like terms. For multiplication, you often use the distributive property, commonly visualized with the FOIL method (First, Outer, Inner, Last) for binomials.

Multiply $(2x+1)(x-3)$ using FOIL:

• First: $2x\cdot x=2x^2$
• Outer: $2x\cdot (-3)=-6x$
• Inner: $1\cdot x=x$
• Last: $1\cdot (-3)=-3$

Combine the middle like terms: $2x^2+(-6x+x)-3=2x^2-5x-3$.

Translating Word Problems into Equations

This is often the most challenging skill. The key is to translate the English sentence, piece by piece, into algebraic components.

1. Identify the unknown and assign a variable.
2. Look for key phrases: "sum" implies addition, "less than" implies subtraction, "product" implies multiplication, "quotient" implies division, "is" or "was" implies equals.
3. Pay close attention to order, especially with subtraction and division ("5 less than x" is $x-5$, not $5-x$).

Example: "Three times a number, decreased by eight, is equal to twice the same number increased by four."

1. Let the number be $n$.
2. "Three times a number" is $3n$. "Decreased by eight" is $-8$. So the first part is $3n-8$.
3. "Is equal to" means $=$.
4. "Twice the same number" is $2n$. "Increased by four" is $+4$. So the second part is $2n+4$.
5. The equation is $3n-8=2n+4$. Solving gives $n=12$.

Common Pitfalls

1. Misapplying the Distributive Property: A common error is to forget to multiply every term inside the parentheses. Incorrect: $4(x-2)=4x-2$. Correct: $4(x-2)=4x-8$. This also applies to negative signs: $-3(2x-5)=-6x+15$, not $-6x-15$.

1. Forgetting to Flip the Inequality Sign: The most frequent inequality mistake. Always remember that multiplying or dividing by a negative number reverses the direction of the inequality symbol. Treat this step as a mandatory checkpoint.

1. Combining Unlike Terms: You cannot combine terms with different variables or exponents. $4x+5y$ cannot become $9xy$, and $x^2+x$ remains $x^2+x$. These are not like terms.

1. Exponent Rule Confusion: Do not multiply the bases when using the product rule. $x^2\cdot x^3=x^5$, not $x^6$. Also, the power rule applies to the entire base: $(2x^3{)}^2=4x^6$, not $2x^6$ (you must square the coefficient 2 as well).

Summary

• Elementary algebra is a significant, foundational portion of the ACT Math test, focusing on linear equations, inequalities, expressions, exponents, and polynomials.
• Success hinges on a flawless execution of core procedures: simplifying expressions, isolating variables in equations, and correctly applying exponent rules.
• The critical exception to remember is that multiplying or dividing an inequality by a negative number requires you to reverse the inequality symbol.
• Translating word problems requires a careful, word-by-word approach to convert English phrases into accurate algebraic equations.
• Avoid common traps like improper distribution, miscombining terms, and mixing up exponent rules by practicing these routines until they are automatic.