# Equations and Inequalities

## Linear Equations in One Variable


### Learning Objectives

1. Simplify expressions using order of operations (IA 1.1.3)
2. Solve linear equations using a general strategy (IA 2.1.1)

### Objective 1: Simplify expressions using order of operations (IA 1.1.3)


### Practice Makes Perfect

Evaluate the following expressions being sure to follow the order of operations:
Simplify by combining like terms:

### Objective 2: Solve linear equations using a general strategy (IA 2.1.1)


### Practice Makes Perfect

Solve each linear equation using the general strategy. 
Caroline is a full-time college student planning a spring break vacation. To earn enough money for the trip, she has taken a part-time job at the local bank that pays $15.00/hr, and she opened a savings account with an initial deposit of $400 on January 15. She arranged for direct deposit of her payroll checks. If spring break begins March 20 and the trip will cost approximately $2,500, how many hours will she have to work to earn enough to pay for her vacation? If she can only work 4 hours per day, how many days per week will she have to work? How many weeks will it take? In this section, we will investigate problems like this and others, which generate graphs like the line in .

### Solving Linear Equations in One Variable

A linear equation is an equation of a straight line, written in one variable. The only power of the variable is 1. Linear equations in one variable may take the form 
  and are solved using basic algebraic operations.
We begin by classifying linear equations in one variable as one of three types: identity, conditional, or inconsistent. An identity equation is true for all values of the variable. Here is an example of an identity equation.
The solution set consists of all values that make the equation true. For this equation, the solution set is all real numbers because any real number substituted for 
  will make the equation true.
A conditional equation is true for only some values of the variable. For example, if we are to solve the equation 
  we have the following:
The solution set consists of one number: 
  It is the only solution and, therefore, we have solved a conditional equation.
An inconsistent equation results in a false statement. For example, if we are to solve 
  we have the following:
Indeed, 
  There is no solution because this is an inconsistent equation.
Solving linear equations in one variable involves the fundamental properties of equality and basic algebraic operations. A brief review of those operations follows.

### Solving a Rational Equation

In this section, we look at rational equations that, after some manipulation, result in a linear equation. If an equation contains at least one rational expression, it is a considered a rational equation.
Recall that a rational number is the ratio of two numbers, such as 
  or 
  A rational expression is the ratio, or quotient, of two polynomials. Here are three examples.
Rational equations have a variable in the denominator in at least one of the terms.
Our goal is to perform algebraic operations so that the variables appear in the numerator. In fact, we will eliminate all denominators by multiplying both sides of the equation by the least common denominator (LCD).
Finding the LCD is identifying an expression that contains the highest power of all of the factors in all of the denominators. We do this because when the equation is multiplied by the LCD, the common factors in the LCD and in each denominator will equal one and will cancel out.
A common mistake made when solving rational equations involves finding the LCD when one of the denominators is a binomial—two terms added or subtracted—such as 
  Always consider a binomial as an individual factor—the terms cannot be separated. For example, suppose a problem has three terms and the denominators are 
  
  and 
  First, factor all denominators. We then have 
  
  and 
  as the denominators. (Note the parentheses placed around the second denominator.) Only the last two denominators have a common factor of 
  The 
  in the first denominator is separate from the 
  in the 
  denominators. An effective way to remember this is to write factored and binomial denominators in parentheses, and consider each parentheses as a separate unit or a separate factor. The LCD in this instance is found by multiplying together the 
  one factor of 
  and the 3. Thus, the LCD is the following:
So, both sides of the equation would be multiplied by 
  Leave the LCD in factored form, as this makes it easier to see how each denominator in the problem cancels out.
Another example is a problem with two denominators, such as 
  and 
  Once the second denominator is factored as 
  there is a common factor of x in both denominators and the LCD is 
  
Sometimes we have a rational equation in the form of a proportion; that is, when one fraction equals another fraction and there are no other terms in the equation.
We can use another method of solving the equation without finding the LCD: cross-multiplication. We multiply terms by crossing over the equal sign.
Multiply 
  and 
  which results in 
  
Any solution that makes a denominator in the original expression equal zero must be excluded from the possibilities.

### Finding a Linear Equation

Perhaps the most familiar form of a linear equation is the slope-intercept form, written as 
  where 
  and 
  Let us begin with the slope.

### The Slope of a Line 

The slope of a line refers to the ratio of the vertical change in y over the horizontal change in x between any two points on a line. It indicates the direction in which a line slants as well as its steepness. Slope is sometimes described as rise over run.
If the slope is positive, the line slants to the right. If the slope is negative, the line slants to the left. As the slope increases, the line becomes steeper. Some examples are shown in . The lines indicate the following slopes: 
  
  and 
  

### The Point-Slope Formula

Given the slope and one point on a line, we can find the equation of the line using the point-slope formula.
This is an important formula, as it will be used in other areas of college algebra and often in calculus to find the equation of a tangent line. We need only one point and the slope of the line to use the formula. After substituting the slope and the coordinates of one point into the formula, we simplify it and write it in slope-intercept form.

### Standard Form of a Line

Another way that we can represent the equation of a line is in standard form. Standard form is given as
where 
  
  and 
  are integers. The x- and y-terms are on one side of the equal sign and the constant term is on the other side.

### Vertical and Horizontal Lines

The equations of vertical and horizontal lines do not require any of the preceding formulas, although we can use the formulas to prove that the equations are correct. The equation of a vertical line is given as
where c is a constant. The slope of a vertical line is undefined, and regardless of the y-value of any point on the line, the x-coordinate of the point will be c.
Suppose that we want to find the equation of a line containing the following points: 
  and 
  First, we will find the slope.
Zero in the denominator means that the slope is undefined and, therefore, we cannot use the point-slope formula. However, we can plot the points. Notice that all of the x-coordinates are the same and we find a vertical line through 
  See .
The equation of a horizontal line is given as
where c is a constant. The slope of a horizontal line is zero, and for any x-value of a point on the line, the y-coordinate will be c.
Suppose we want to find the equation of a line that contains the following set of points: 
  and 
  We can use the point-slope formula. First, we find the slope using any two points on the line.
Use any point for 
  in the formula, or use the y-intercept. 
The graph is a horizontal line through 
  Notice that all of the y-coordinates are the same. See .

### Determining Whether Graphs of Lines are Parallel or Perpendicular

Parallel lines have the same slope and different y-intercepts. Lines that are parallel to each other will never intersect. For example,  shows the graphs of various lines with the same slope, 
  
All of the lines shown in the graph are parallel because they have the same slope and different y-intercepts.
Lines that are perpendicular intersect to form a 
  -angle. The slope of one line is the negative reciprocal of the other. We can show that two lines are perpendicular if the product of the two slopes is 
  For example,  shows the graph of two perpendicular lines. One line has a slope of 3; the other line has a slope of 
  

### Writing the Equations of Lines Parallel or Perpendicular to a Given Line

As we have learned, determining whether two lines are parallel or perpendicular is a matter of finding the slopes. To write the equation of a line parallel or perpendicular to another line, we follow the same principles as we do for finding the equation of any line. After finding the slope, use the point-slope formula to write the equation of the new line.

### Key Concepts

1. We can solve linear equations in one variable in the form 
  using standard algebraic properties. See  and .
2. A rational expression is a quotient of two polynomials. We use the LCD to clear the fractions from an equation. See  and .
3. All solutions to a rational equation should be verified within the original equation to avoid an undefined term, or zero in the denominator. See  and  and .
4. Given two points, we can find the slope of a line using the slope formula. See .
5. We can identify the slope and y-intercept of an equation in slope-intercept form. See .
6. We can find the equation of a line given the slope and a point. See . 
7. We can also find the equation of a line given two points. Find the slope and use the point-slope formula. See .
8. The standard form of a line has no fractions. See .
9. Horizontal lines have a slope of zero and are defined as 
  where c is a constant.
10. Vertical lines have an undefined slope (zero in the denominator), and are defined as 
  where c is a constant. See .
11. Parallel lines have the same slope and different y-intercepts. See  and .
12. Perpendicular lines have slopes that are negative reciprocals of each other unless one is horizontal and the other is vertical. See .

### Section Exercises


### Verbal


### Algebraic

For the following exercises, solve the equation for 
  
For the following exercises, solve each rational equation for 
  State all x-values that are excluded from the solution set.
For the following exercises, find the equation of the line using the point-slope formula.
Write all the final equations using the slope-intercept form.
For the following exercises, find the equation of the line using the given information.

### Graphical

For the following exercises, graph the pair of equations on the same axes, and state whether they are parallel, perpendicular, or neither.

### Numeric

For the following exercises, find the slope of the line that passes through the given points.
For the following exercises, find the slope of the lines that pass through each pair of points and determine whether the lines are parallel or perpendicular.

### Technology

For the following exercises, express the equations in slope intercept form (rounding each number to the thousandths place). Enter this into a graphing calculator as Y1, then adjust the ymin and ymax values for your window to include where the y-intercept occurs. State your ymin and ymax values.

### Extensions


### Real-World Applications

For the following exercises, use this scenario: The cost of renting a car is $45/wk plus $0.25/mi traveled during that week. An equation to represent the cost would be 
  where 
  is the number of miles traveled.