Algebraic Expressions, Equations, and Inequalities

Algebraic Equations

Writing Equations

Algebraic equations can represent verbal statements of situations.

An equation is a number sentence stating that two expressions have the same value. The difference between an expression and an equation is the equal sign, ==. If a problem is stated verbally, the first step in solving is to translate the description of the problem into an algebraic equation.

Some words and phrases that represent the equal sign == include:

  • Amounts to
  • Equals
  • Gives
  • Is
  • Is the same as
  • Represents
  • Yields

To translate a problem description into an equation, look for the word or phrase that represents the equal sign. Determine what part of the problem describes the expression on each side. Translate the two expressions, and place the equal sign between them.

Writing Algebraic Equations

Description Equation
The sum of a number and 32\text{{\color{#c42126}{The sum of a number and 32}}} is 54{\color{#0047af}{54}}.
n+32=54{\color{#c42126}{n+32}}={\color{#0047af}{54}}
The cost of 5 pounds of bananas\text{{\color{#c42126}{The cost of 5 pounds of bananas}}} is $3.25{\color{#0047af}{\$3.25}}.
5p=3.25{\color{#c42126}{5p}}={\color{#0047af}{3.25}}
Gabriela has earned 117 credit hours, which\text{{\color{#c42126}{Gabriela has earned 117 credit hours, which}}} is 3 fewer than the number of credits she needs to graduate\text{{\color{#0047af}{3 fewer than the number of credits she needs to graduate}}}.
117=c3{\color{#c42126}{117}}={\color{#0047af}{c-3}}

Step-By-Step Example
Writing Equations from Word Problems

A library has 86 science fiction paperbacks and e-books combined. The number of paperbacks are 4 fewer than twice the number of e-books.

Write an equation that represents the combined number of science fiction paperbacks and e-books.

Step 1

Choose a variable.

The number of paperbacks is described in relation to the number of e-books. So, let xx represent the number of e-books:
x=Number of e-booksx=\text{Number of e-books}
Step 2
Write an expression for the number of paperbacks. There are 4 fewer paperbacks than twice the number of e-books. The phrase twice the number means to multiply by 2, or 2x2x. The phrase 4 fewer means to subtract 4. So, the number of paperbacks is:
2x4=Number of paperbacks2x-4=\text{Number of paperbacks}
Step 3
Write an equation that represents the combined number of paperbacks and e-books. The library has 86 science fiction books combined. 86 is the total number of science fiction books. The total includes the number of paperbacks, or 2x42x-4. It also includes the number of e-books, or xx. The word has represents the equal sign. The word combined represents addition. So, add the expressions that represent the number of paperbacks and e-books:
86=Paperbacks+E-books86=(2x4)+x\begin{aligned}86&=\text{Paperbacks}+\text{E-books}\\86&=(2x-4)+x\end{aligned}
Step 4
Simplify the equation.
86=(2x4)+x86=2x4+x86=3x4\begin{aligned}86&=(2x-4)+x\\86&=2x-4+x\\86&=3x-4\end{aligned}
Solution
The equation for the combined number of science fiction paperbacks and e-books is:
86=3x486=3x-4
The variable xx represents the number of e-books.

Application of Equations

Equations have many applications that aid in calculating values that are used in the real world.
Algebraic equations are used in the real world to determine values that are needed for calculations. Formulas are algebraic equations that represent real-world quantities. Some formulas are used to find measurements, such as perimeter and area, while other formulas are used in financial calculations, such as determining the amount of money in an account that earns monthly interest after a number of years.

Geometric Formulas

Formulas are a form of algebraic equations. Common formulas, like the perimeter (PP) and area (AA) of a rectangle and triangle, use variables to represent different values, such as the base (bb) and height (hh) of each shape.
Step-By-Step Example
Calculating Geometric Formulas
Find the perimeter and area of a rectangle with a base of 8 centimeters (cm) and a height of 15 cm.
Step 1
Determine the appropriate formulas.
Perimeter (P)=2b+2hArea (A)=bh\text{Perimeter }(P)=2b+2h \;\;\;\;\;\;\;\;\;\;\text{Area }(A)=bh
Step 2
Identify the values of the variables. The base, bb, is 8 cm. The height, hh, is 15 cm.
Step 3
Substitute the given dimensions into the formulas.
P=2b+2hP=2(8)+2(15)A=bhA=8(15)\begin{aligned}P&=2b+2h \\ P&=2(8)+2(15)\end{aligned} \;\;\;\;\;\;\;\;\;\; \begin{aligned}A&=bh \\ A&=8(15)\end{aligned}
Step 4
Simplify.
P=16+30P=46A=8(15)A=120\begin{aligned}P&=16+30\\P&=46\end{aligned}\;\;\;\;\;\;\;\;\;\;\begin{aligned}A&=8(15)\\A&=120\end{aligned}
Solution

Write the values using the appropriate units of measure.

The perimeter is 46 cm.

The area is 120 square centimeters.

Step-By-Step Example
Solving a Financial Formula

Eve opens an account with $100. The account earns 6% interest compounded monthly. How much will be in the account after 5 years?

Use the formula for compounded interest, or interest that is added back into the amount that earns interest.
A=P(1+r12)12tA=P\left(1+\frac{r}{12}\right)^{12t}
The amount of money in the account is AA. The principal, or original amount, is PP. The interest rate is rr. The number of years is tt.
Step 1
Use the formula for compounded interest:
A=P(1+r12)12tA=P\left(1+\frac{r}{12}\right)^{12t}
Identify the values of the variables.

The original amount in the account is $100, or PP.

The rate, rr, of 6% expressed as a decimal is 0.06.

The time, tt, is years.

Step 2
Substitute the values of the variables into the formula.
A=100(1+0.0612)125A=100\left(1+\frac{0.06}{12}\right)^{12\cdot5}
Step 3
Simplify according to the order of operations. First, simplify within the parentheses.
A=100(1+0.005)125A=100(1.005)125\begin{aligned}A&=100\left(1+0.005\right)^{12\cdot5}\\A&=100(1.005)^{12\cdot5}\end{aligned}
Step 4
The expression 12512\cdot5 is grouped in the exponent. Perform the multiplication and then simplify the exponent.
A=100(1.005)60A100(1.34885015)\begin{aligned}A&=100\left(1.005\right)^{60}\\A&\approx100(1.34885015)\end{aligned}
Avoid rounding until the end of the problem to reduce rounding errors.
Solution
Multiply. The value is in dollars. So, round to the nearest hundredth, or cent.
A134.89A\approx134.89
After 5 years, there will be $134.89 in the account.

Checking Solutions of Equations

To check a solution of an equation, substitute the value of the variables in the equation, and determine whether the resulting statement is true.
An equation is true if both sides are equal. Solutions of an equation make the equation true. For example, x=2x=2 is a solution of the equation:
x+2=4x+2=4
However, x=3x=3 is not a solution. To determine whether a value of the variable is a solution of an equation, substitute the value into the equation and evaluate both sides. If they are equal, then it is a solution.
To check several solutions of an equation, substitute the values for the variable and simplify to determine whether the resulting equations are true.