Open Resources for Community College Algebra

A variable represents an unknown quantity, or a quantity that can change. Algebra often uses \(x\) as the variable, but any letter or word can work as a variable. We also often use a letter that stands for something, like \(g\) for gas mileage.

When a variable represents a physical quantity, be clear about which units of measurement apply. For example, if \(g\) measures gas mileage in miles per gallon, that is different from \(g\) measuring gas mileage in liters per kilometer.

An algebraic expression is any combination of variables and numbers using arithmetic operations like addition, multiplication, etc. Algebraic expressions can be evaluated. This means substituting values in for the variable(s).

Be careful when evaluating an algebraic expression at a negative number. It helps to wrap parentheses around any negative number you substitute in.

In an algebraic expression, terms are pieces of the expression that are added together. For example, the terms in \(2x^2-5x+7\) are \(2x^2\text{,}\) \(-5x\text{,}\) and \(7\text{.}\)

Terms are different from factors, which are pieces of an algebraic expression that are multiplied together. For example, the factors of \(2x(x+5)\) are \(2\text{,}\) \(x\text{,}\) and \((x+5)\text{.}\)

Like terms arise in applications where it makes sense to add some things together, and it might happen that the terms you have to add are similar enough to be called like terms. For example, finding a perimeter of a polygon might be an application of combining like terms, if the sides of the polygon are each labeled as a number times some common variable.

The symbols used for comparing two quantities are as follows:

An interval is a collection of numbers on a number line that are all connected. We illustrate intervals with a number line, where some portion of the number line is shaded. To clear up whether or not an endpoint of the shaded region is part off the interval, we use brackets (to include that number) or parentheses (to exclude that number). An interval might extend forever in one direction, and then the graph uses an arrowhead. For example, here is a graph of the interval of all positive numbers.

And here is an interval with all numbers that are less than or equal to \(4\text{.}\)

The other standard notation for an interval is interval notation. This notation identifies the left and right ends of an interval and just writes them down, separated by a comma. Brackets or parentheses indicate whether the end is included or not in the interval. For example, \((0,\infty)\) is the interval of all positive numbers, and \([0,\infty)\) is the interval of all non-negative numbers (meaning the positive numbers and also zero).

An equation is a statement that two algebraic expressions are equal. There must be an equal sign (\(=\)) in between the two expressions. For example, \(x^2=x+2\) in an equation. An inequality is similar, but uses one of the five inequality symbols instead of an equal sign. An inequality is a statement about how the two expressions relate to each other.

When an equation or inequality only has one variable, a solution to the equation or inequality is a number that you can substitute in for the variable and it results in a true relation between pure numbers. For example, \(2\) is a solution to \(x^2=x+2\) because when you substitute \(2\) in for \(x\) and simplify each side, you have \(4\confirm{=}4\text{.}\) But for example, \(3\) is not a solution, since when you substitute \(3\) in for \(x\) and simplify each side, you have \(9\reject{=}5\text{.}\) The skill of checking whether or not a given number is a solution to an equation or inequality is important.

A linear expression in one variable is an expression that simplifies to the form \(ax+b\) where \(a\) and \(b\) are specific numbers, but \(a\neq0\text{.}\) For example, \(\frac{2}{5}x+3\) and \(5(x+3)-2x\) are linear expressions.

A linear equation is a specific type of equation where the two sides of the equation are either both linear expressions, or one side is a linear expression and the other side is just a number. A linear inequality is similar but it’s an inequality, not an equation. Linear equations and inequalities are the focus of Part I of this textbook series.

Suppose you would like to find the solution(s) to an equation like \(x+7=11\text{.}\) There is a formal process we can follow to do this. It converts \(x+7=11\) into an equivalent equation (an equation with the same solution set). Specifically, we have a process that isolates \(x\text{,}\) leaving us with an equivalent equation that directly states what \(x\) must equal.

Since the variable has \(7\) added to it, we must take the opposite action, subtracting \(7\text{.}\) And we must do that to both sides, not just the left side. After doing that we have the equivalent equation \(x=4\text{.}\) So the only solution is \(4\text{.}\)

Adding and subtracting are opposite operations. Multiplying and dividing are opposite operations. Keeping these pairings of opposite actions in mind, we can solve many small linear equations. According to Fact 1.5.12, we can always add or subtract any number on each side of an equation to obtain an equivalent equation. And we can always multiply or divide by any nonzero number to obtain an equivalent equation.

If a variable that you need to isolate is being multiplied by a fraction, then multiplying by the reciprocal of that fraction is one way to undo that. Of course, this is still an action that you must take to both sides of the equation.

The solution set to an equation is the collection of all numbers that are solutions. For the linear equation \(x+7=11\text{,}\) there was only one solution, so the solution set is a “collection” that only has one number in it. Whenever a solution set only has a finite number of numbers in it, we use braces to write the solution set. In this case, the solution set is \(\{4\}\text{.}\) This is called set notation (not to be confused with set-builder notation).

Solving linear inequalities is a lot like solving linear equations, but there are two important differences. One difference is that typically, the solution set is an interval of numbers. So it can be expressed using a number line graph, interval notation, or set-builder notation.

Also, whenever the solving process requires you to multiply or divide on each side by a negative number, the direction of the inequality symbol changes. For example when solving \(-2x\leq24\text{,}\) we would divide on each side by \(-2\text{.}\) And then we would have to change the direction of the inequality symbol and end with \(x\geq-12\text{.}\)

The number \(0\) is called the additive identity because you can add \(0\) to any number and the value does not change. A number’s additive inverse (or opposite) is the number you can add to it to get \(0\text{.}\) In other words, its negative. For example the additive inverse of \(8\) is \(-8\text{,}\) and the additive inverse of \(-3.4\) is \(3.4\text{.}\)

The number \(1\) is called the multiplicative identity because you can multiply any number by \(1\) and the value does not change. A number’s multiplicative inverse (or reciprocal) is the number you can multiply it by to get \(1\text{.}\) For example the multiplicative inverse of \(6\) is \(\frac{1}{6}\text{,}\) and the multiplicative inverse of \(-\frac{3}{2}\) is \(-\frac{2}{3}\text{.}\)

A commutative property allows you to write two numbers or expressions in the opposite order and have an equal result. For example, \(8+3 = 3+8\text{.}\) This illustrates that addition has the commutative property. And for example, \(4(9) = 9(4)\text{.}\) This illustrates that multiplication has the commutative property. Note that subtraction and division do not have the commutative property.

An associative property allows you to group three numbers or expressions in a different way without changing the order they are written. For example, \((8+3)+5 = 8+(3+5)\text{.}\) This illustrates that addition has the associative property. And for example, \((2\cdot3)(5) = 2(3\cdot5)\text{.}\) This illustrates that multiplication has the associative property. Note that subtraction and division do not have the associative property.

Technically, all these concepts above are the reasons why we can do things like combine like terms and simplify many kinds of algebra expressions. We learn about these concepts here, and yet you might find that you don’t need to literally use them to succeed with solving algebra problems.

When you have a “word problem” in front of you, the first thing to do is read and re-read everything until you have an understanding of what the numbers really represent physically, and an understanding of what exactly you are being asked to find. Once you have that understanding, clearly define a variable that represents whatever quantity you need to find. And clearly state what units of measure go with that variable, if there are any.

In this section, we are concerned with setting up these equations. Later we will actually solve them and answer the underlying application question. But it is challenging enough for now just to correctly set up these equations.

Occasionally, it is more appropriate to set up an inequality than an equation. Look for phrases like “is at most”, “needs to be at least”, etc. And look for words that imply these meanings, like “maximum”, “minimum”, etc. And use reading comprehension to understand when this is implied. For example if a person is working with a budget, they are required to spend no more than that amount. They could spend it all, or spend less.