ACT Math 2: Elementary Algebra — Linear Equations and Inequalities

Constant

A number whose value remains fixed in a particular expression or relationship. For example, the 3 in $g+3$ is a constant.

How can Greg's age be represented by $g$ if Alex is always 3 years older?

Greg's age is $g$, so Alex's age is represented by $g+3$. The variable changes with age, while the difference of 3 remains constant.

What algebraic expression represents the sum of $a$ and $b$?

$a+b$.

What algebraic expression represents the difference of $a$ and $b$?

$a-b$, meaning $a$ minus $b$. The order matters in subtraction.

What algebraic forms represent the product of $a$ and $b$?

$a\cdot b$, $ab$, $(a)(b)$, or $a(b)$. The multiplication symbol $\times$ is generally avoided in algebra because it can be confused with a variable.

What algebraic forms represent the quotient of $a$ and $b$?

$a\div b$, $a/b$, or $\frac{a}{b}$. Here, $a$ is the dividend and $b$ is the divisor.

How should the inequality $a<b$ be interpreted, and what equivalent reversed form expresses the same relationship?

$a$ is less than $b$, meaning $a$ lies to the left of $b$ on a number line. The equivalent reversed form is $b>a$.

What do the symbols $\ne$, $<$, $\le$, $>$, and $\ge$ mean?

$\ne$ means not equal to; $<$ means less than; $\le$ means less than or equal to; $>$ means greater than; and $\ge$ means greater than or equal to.

Grouping symbols

Parentheses $( )$, brackets $[ ]$, and braces $\{\}$ indicate which parts of an expression should be treated together. With nested grouping, simplify the innermost group first.

Expression

A number, variable, or combination of numbers and variables connected by operation symbols. An expression does not contain an equal sign, such as $4(y-1)+1$.

Equation

A statement that two expressions are equal and are connected by an equal sign, such as $2(x+3)=10$.

In exponential notation $a^n$, what are the base and exponent, and what does the notation mean?

$a$ is the base and $n$ is the exponent. For a positive integer $n$, $a^n$ means multiplying $a$ by itself $n$ times.

What are the special names for $a^2$ and $a^3$?

$a^2$ is read as $a$ squared, and $a^3$ is read as $a$ cubed.

What is the expanded form of $2^4$?

$2\cdot2\cdot2\cdot2$.

Simplify $3^4$.

$3^4=3\cdot3\cdot3\cdot3=81$.

Order of operations

Simplify grouping symbols first, then exponents, then multiplication and division from left to right, and finally addition and subtraction from left to right.

Why are multiplication and division performed from left to right rather than always doing multiplication first?

Multiplication and division have equal priority. For example, $12\div3\cdot2$ is evaluated as $(12\div3)\cdot2=8$, not $12\div(3\cdot2)$.

Why are addition and subtraction performed from left to right?

Addition and subtraction have equal priority. For example, $10-3+2$ is evaluated as $(10-3)+2=9$.

Simplify $4+3\cdot7$ and $(4+3)\cdot7$.

$4+3\cdot7=4+21=25$, whereas $(4+3)\cdot7=7\cdot7=49$. Grouping symbols change the order of operations.

How should an expression with nested grouping symbols be simplified?

Start with the innermost parentheses or grouping symbols, simplify outward, then apply exponents, multiplication or division, and addition or subtraction according to the order of operations.

Simplify $18\div6+4(5-2)$.

First evaluate the parentheses: $5-2=3$. Then divide and multiply from left to right: $18\div6+4(3)=3+12=15$.

Evaluate an expression

To evaluate an expression, substitute the given value for each variable and then simplify using the order of operations.

Evaluate $2x^2+3x+8$ when $x=4$.

Substitute $x=4$: $2(4^2)+3(4)+8=2(16)+12+8=52$.

What is the difference between evaluating and simplifying an algebraic expression?

Simplifying performs all possible operations without assigning values to variables. Evaluating substitutes specified numerical values and then simplifies.

Term

A constant or a product of a constant and one or more variables, separated from other terms by addition or subtraction. Examples include $7$, $y$, $5x^2$, and $9a$.

Coefficient

The numerical factor multiplying a variable in a term. The coefficient of $3x$ is 3; the coefficient of $x$ is 1 because $x=1x$.

Like terms

Terms that are constants or have exactly the same variables raised to exactly the same powers. For example, $5x^2$ and $9x^2$ are like terms, but $x$ and $x^2$ are not.

How do you combine like terms?

Identify terms with matching variable parts, group them if useful, add or subtract their coefficients, and retain the common variable part. For example, $4x+7x+x=(4+7+1)x=12x$.

Simplify $2x^2+3x+7+x^2+4x+5$.

Combine matching terms: $(2+1)x^2+(3+4)x+(7+5)=3x^2+7x+12$.

Why cannot $3x$ and $4x^2$ be combined as like terms?

Their variable parts are different: one contains $x^1$ and the other contains $x^2$. Only terms with identical variables and exponents can be combined directly.

Translate “the difference of $17x$ and 5” into an algebraic expression.

$17x-5$.

Translate “the quotient of $10x^2$ and 7” into an algebraic expression.

$\frac{10x^2}{7}$.

Translate “eleven more than $x$” and “fourteen less than $11a$.”

They translate to $x+11$ and $11a-14$, respectively. In “less than” phrases, the stated amount is subtracted from the quantity that follows “than.”

Translate “five times the sum of $m$ and $n$” into an algebraic expression.

$5(m+n)$. Parentheses are required because the sum is formed before multiplying by 5.

How does “the sum of five times $m$ and $n$” differ from “five times the sum of $m$ and $n$”?

The sum of five times $m$ and $n$ is $5m+n$. Five times the sum of $m$ and $n$ is $5(m+n)$.

Translate “the difference of two times $x$ and 8” and “two times the difference of $x$ and 8.”

The first is $2x-8$; the second is $2(x-8)$. The placement of “two times” determines whether the factor applies to the entire difference.

A rectangle's width is 6 less than its length. If the length is $l$, what expression represents the width?

$l-6$. “Six less than the length” means 6 is subtracted from $l$.

The number of dimes is three less than four times the number of quarters. If $q$ is the number of quarters, what expression represents the number of dimes?

$4q-3$.