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# Linear Inequalities and Absolute Value Inequalities

It is not easy to make the honor role 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.
Figure 1

## Write and Manipulate Inequalities

Indicating the solution to an inequality such as $x\ge 4$ can be achieved in several ways. We can use a number line as shown below. The blue ray begins at $x=4$ 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: $\{x|x\ge 4\}$, 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 $x\ge 4$ are represented as $\left[4,\infty \right)$. 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 $\left[-2,6\right)$, or all numbers between $-2$ and $6$, including $-2$, but not including $6$; $\left(-1,0\right)$, all real numbers between, but not including $-1$ and $0$; and $\left(-\infty ,1\right]$, all real numbers less than and including $1$. The table below outlines the possibilities.
Set Indicated Set-Builder Notation Interval Notation
All real numbers between a and b, but not including a or b $\{x|a<x<b\}$ $\left(a,b\right)$
All real numbers greater than a, but not including a $\{x|x>a\}$ $\left(a,\infty \right)$
All real numbers less than b, but not including b $\{x|x<b\}$ $\left(-\infty ,b\right)$
All real numbers greater than a, including a $\{x|x\ge a\}$ $\left[a,\infty \right)$
All real numbers less than b, including b $\{x|x\le b\}$ $\left(-\infty ,b\right]$
All real numbers between a and b, including a $\{x|a\le x<b\}$ $\left[a,b\right)$
All real numbers between a and b, including b $\{x|a<x\le b\}$ $\left(a,b\right]$
All real numbers between a and b, including a and b $\{x|a\le x\le b\}$ $\left[a,b\right]$
All real numbers less than a or greater than b $\{x|x<a\text{ and }x>b\}$ $\left(-\infty ,a\right)\cup \left(b,\infty \right)$
All real numbers $\{x|x\text{ is all real numbers}\}$ $\left(-\infty ,\infty \right)$

### Example: Using Interval Notation to Express All Real Numbers Greater Than or Equal to a

Use interval notation to indicate all real numbers greater than or equal to $-2$.

Answer: Use a bracket on the left of $-2$ and parentheses after infinity: $\left[-2,\infty \right)$. The bracket indicates that $-2$ is included in the set with all real numbers greater than $-2$ to infinity.

### Try It

Use interval notation to indicate all real numbers between and including $-3$ and $5$.

### Example: Using Interval Notation to Express All Real Numbers Less Than or Equal to a or Greater Than or Equal to b

Write the interval expressing all real numbers less than or equal to $-1$ or greater than or equal to $1$.

Answer: We have to write two intervals for this example. The first interval must indicate all real numbers less than or equal to 1. So, this interval begins at $-\infty$ and ends at $-1$, which is written as $\left(-\infty ,-1\right]$. The second interval must show all real numbers greater than or equal to $1$, which is written as $\left[1,\infty \right)$. However, we want to combine these two sets. We accomplish this by inserting the union symbol, $\cup$, between the two intervals.

$\left(-\infty ,-1\right]\cup \left[1,\infty \right)$

### Try It

Express all real numbers less than $-2$ or greater than or equal to 3 in interval notation.

### Example: Demonstrating the Multiplication Property

Illustrate the multiplication property for inequalities by solving each of the following:
1. $3x<6$
2. $-2x - 1\ge 5$
3. $5-x>10$

1. $\begin{array}{l}3x<6\hfill \\ \frac{1}{3}\left(3x\right)<\left(6\right)\frac{1}{3}\hfill \\ x<2\hfill \end{array}$
2. $\begin{array}{ll}-2x - 1\ge 5\hfill & \hfill \\ -2x\ge 6\hfill & \hfill \\ \left(-\frac{1}{2}\right)\left(-2x\right)\ge \left(6\right)\left(-\frac{1}{2}\right)\hfill & \text{Multiply by }-\frac{1}{2}.\hfill \\ x\le -3\hfill & \text{Reverse the inequality}.\hfill \end{array}$
3. $\begin{array}{ll}5-x>10\hfill & \hfill \\ -x>5\hfill & \hfill \\ \left(-1\right)\left(-x\right)>\left(5\right)\left(-1\right)\hfill & \text{Multiply by }-1.\hfill \\ x<-5\hfill & \text{Reverse the inequality}.\hfill \end{array}$

### Try It

Solve $4x+7\ge 2x - 3$.

Answer: $x\ge -5$

### Example: Solving an Inequality with Fractions

Solve the following inequality and write the answer in interval notation: $-\frac{3}{4}x\ge -\frac{5}{8}+\frac{2}{3}x$.

Answer: We begin solving in the same way we do when solving an equation.

$\begin{array}{ll}-\frac{3}{4}x\ge -\frac{5}{8}+\frac{2}{3}x\hfill & \hfill \\ -\frac{3}{4}x-\frac{2}{3}x\ge -\frac{5}{8}\hfill & \text{Put variable terms on one side}.\hfill \\ -\frac{9}{12}x-\frac{8}{12}x\ge -\frac{5}{8}\hfill & \text{Write fractions with common denominator}.\hfill \\ -\frac{17}{12}x\ge -\frac{5}{8}\hfill & \hfill \\ x\le -\frac{5}{8}\left(-\frac{12}{17}\right)\hfill & \text{Multiplying by a negative number reverses the inequality}.\hfill \\ x\le \frac{15}{34}\hfill & \hfill \end{array}$
The solution set is the interval $\left(-\infty ,\frac{15}{34}\right]$.

### Try It

Solve the inequality and write the answer in interval notation: $-\frac{5}{6}x\le \frac{3}{4}+\frac{8}{3}x$.

### Example: Solving a Compound Inequality with the Variable in All Three Parts

Solve the compound inequality with variables in all three parts: $3+x>7x - 2>5x - 10$.

Answer: Lets try the first method. Write two inequalities:

$\begin{array}{lll}3+x> 7x - 2\hfill & \text{and}\hfill & 7x - 2> 5x - 10\hfill \\ 3> 6x - 2\hfill & \hfill & 2x - 2> -10\hfill \\ 5> 6x\hfill & \hfill & 2x> -8\hfill \\ \frac{5}{6}> x\hfill & \hfill & x> -4\hfill \\ x< \frac{5}{6}\hfill & \hfill & -4< x\hfill \end{array}$
The solution set is $-4<x<\frac{5}{6}$ or in interval notation $\left(-4,\frac{5}{6}\right)$. Notice that when we write the solution in interval notation, the smaller number comes first. We read intervals from left to right, as they appear on a number line.

### Try It

Solve the compound inequality: $3y<4 - 5y<5+3y$.

Answer: $\left(-\frac{1}{8},\frac{1}{2}\right)$

## Key Concepts

• 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.
• 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.
• 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.
• 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.
• 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.

## Glossary

compound inequality a problem or a statement that includes two inequalities interval an interval describes a set of numbers within which a solution falls interval notation a mathematical statement that describes a solution set and uses parentheses or brackets to indicate where an interval begins and ends linear inequality similar to a linear equation except that the solutions will include sets of numbers