# Equations and Inequalities

## Linear Inequalities and Absolute Value Inequalities

It is not easy to make the honor roll at most top universities. Suppose students were required to carry a course load of at least 12 credit hours and maintain a grade point average of 3.5 or above. How could these honor roll requirements be expressed mathematically? In this section, we will explore various ways to express different sets of numbers, inequalities, and absolute value inequalities.

### Using Interval Notation

Indicating the solution to an inequality such as 
    can be achieved in several ways.
We can use a number line as shown in . The blue ray begins at 
    and, as indicated by the arrowhead, continues to infinity, which illustrates that the solution set includes all real numbers greater than or equal to 4.
We can use set-builder notation: 
    which translates to “all real numbers x such that x is greater than or equal to 4.” Notice that braces are used to indicate a set.
The third method is interval notation, in which solution sets are indicated with parentheses or brackets. The solutions to 
    are represented as 
    This is perhaps the most useful method, as it applies to concepts studied later in this course and to other higher-level math courses.
The main concept to remember is that parentheses represent solutions greater or less than the number, and brackets represent solutions that are greater than or equal to or less than or equal to the number. Use parentheses to represent infinity or negative infinity, since positive and negative infinity are not numbers in the usual sense of the word and, therefore, cannot be “equaled.” A few examples of an interval, or a set of numbers in which a solution falls, are 
    or all numbers between 
    and 
    including 
    but not including 
    
    all real numbers between, but not including 
    and 
    and 
    all real numbers less than and including 
     outlines the possibilities.

### Using the Properties of Inequalities

When we work with inequalities, we can usually treat them similarly to but not exactly as we treat equalities. We can use the addition property and the multiplication property to help us solve them. The one exception is when we multiply or divide by a negative number; doing so reverses the inequality symbol.

### Solving Inequalities in One Variable Algebraically

As the examples have shown, we can perform the same operations on both sides of an inequality, just as we do with equations; we combine like terms and perform operations. To solve, we isolate the variable.

### Understanding Compound Inequalities

A compound inequality includes two inequalities in one statement. A statement such as 
    means 
    and 
    There are two ways to solve compound inequalities: separating them into two separate inequalities or leaving the compound inequality intact and performing operations on all three parts at the same time. We will illustrate both methods.

### Solving Absolute Value Inequalities

As we know, the absolute value of a quantity is a positive number or zero. From the origin, a point located at 
    has an absolute value of 
    as it is x units away. Consider absolute value as the distance from one point to another point. Regardless of direction, positive or negative, the distance between the two points is represented as a positive number or zero.
An absolute value inequality is an equation of the form
Where A, and sometimes B, represents an algebraic expression dependent on a variable x. Solving the inequality means finding the set of all 
    -values that satisfy the problem. Usually this set will be an interval or the union of two intervals and will include a range of values.
There are two basic approaches to solving absolute value inequalities: graphical and algebraic. The advantage of the graphical approach is we can read the solution by interpreting the graphs of two equations. The advantage of the algebraic approach is that solutions are exact, as precise solutions are sometimes difficult to read from a graph.
Suppose we want to know all possible returns on an investment if we could earn some amount of money within $200 of $600. We can solve algebraically for the set of x-values such that the distance between 
    and 600 is less than or equal to 200. We represent the distance between 
    and 600 as 
    and therefore, 
    or
This means our returns would be between $400 and $800.
To solve absolute value inequalities, just as with absolute value equations, we write two inequalities and then solve them independently.

### Key Concepts

1. Interval notation is a method to indicate the solution set to an inequality. Highly applicable in calculus, it is a system of parentheses and brackets that indicate what numbers are included in a set and whether the endpoints are included as well. See  and .
2. Solving inequalities is similar to solving equations. The same algebraic rules apply, except for one: multiplying or dividing by a negative number reverses the inequality. See , , , and .
3. Compound inequalities often have three parts and can be rewritten as two independent inequalities. Solutions are given by boundary values, which are indicated as a beginning boundary or an ending boundary in the solutions to the two inequalities. See  and .
4. Absolute value inequalities will produce two solution sets due to the nature of absolute value. We solve by writing two equations: one equal to a positive value and one equal to a negative value. See  and .
5. Absolute value inequalities can also be solved by graphing. At least we can check the algebraic solutions by graphing, as we cannot depend on a visual for a precise solution. See .

### Section Exercises


### Verbal


### Algebraic

For the following exercises, solve the inequality. Write your final answer in interval notation.
For the following exercises, solve the inequality involving absolute value. Write your final answer in interval notation.
For the following exercises, describe all the x-values within or including a distance of the given values.
For the following exercises, solve the compound inequality. Express your answer using inequality signs, and then write your answer using interval notation.

### Graphical

For the following exercises, graph the function. Observe the points of intersection and shade the x-axis representing the solution set to the inequality. Show your graph and write your final answer in interval notation.
For the following exercises, graph both straight lines (left-hand side being y1 and right-hand side being y2) on the same axes. Find the point of intersection and solve the inequality by observing where it is true comparing the y-values of the lines.

### Numeric

For the following exercises, write the set in interval notation.
For the following exercises, write the interval in set-builder notation.
For the following exercises, write the set of numbers represented on the number line in interval notation.

### Technology

For the following exercises, input the left-hand side of the inequality as a Y1 graph in your graphing utility. Enter y2 = the right-hand side. Entering the absolute value of an expression is found in the MATH menu, Num, 1:abs(. Find the points of intersection, recall (2nd CALC 5:intersection, 1st curve, enter, 2nd curve, enter, guess, enter). Copy a sketch of the graph and shade the x-axis for your solution set to the inequality. Write final answers in interval notation.

### Extensions


### Real-World Applications


### Chapter Review Exercises


### The Rectangular Coordinate Systems and Graphs

For the following exercises, find the x-intercept and the y-intercept without graphing.
For the following exercises, solve for y in terms of x, putting the equation in slope–intercept form.
For the following exercises, find the distance between the two points.
For the following exercises, find the coordinates of the midpoint of the line segment that joins the two given points.
For the following exercises, construct a table and graph the equation by plotting at least three points.

### Linear Equations in One Variable

For the following exercises, solve for 
    
For the following exercises, solve 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.

### Models and Applications

For the following exercises, write and solve an equation to answer each question.

### Complex Numbers

For the following exercises, use the quadratic equation to solve.
For the following exercises, name the horizontal component and the vertical component.
For the following exercises, perform the operations indicated.

### Quadratic Equations

For the following exercises, solve the quadratic equation by factoring.
For the following exercises, solve the quadratic equation by using the square-root property.
For the following exercises, solve the quadratic equation by completing the square.
For the following exercises, solve the quadratic equation by using the quadratic formula. If the solutions are not real, state No real solution.
For the following exercises, solve the quadratic equation by the method of your choice.

### Other Types of Equations

For the following exercises, solve the equations.

### Linear Inequalities and Absolute Value Inequalities

For the following exercises, solve the inequality. Write your final answer in interval notation.
For the following exercises, solve the compound inequality. Write your answer in interval notation.
For the following exercises, graph as described.

### Chapter Practice Test

For the following exercises, find the equation of the line with the given information.
For the following exercises, find the real solutions of each equation by factoring.