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# Writing and Manipulating Inequalities

### Learning Outcomes

• Use interval notation to express inequalities.
• Use properties of 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, where 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.
Inequality Words Interval Notation
${a}\lt{x}\lt{ b}$ all real numbers between a and b, not including a and b $\left(a,b\right)$
${x}\gt{a}$ All real numbers greater than a, but not including a $\left(a,\infty \right)$
${x}\lt{b}$ All real numbers less than b, but not including b $\left(-\infty ,b\right)$
${x}\ge{a}$ All real numbers greater than a, including a $\left[a,\infty \right)$
${x}\le{b}$ All real numbers less than b, including b $\left(-\infty ,b\right]$
${a}\le{x}\lt{ b}$ All real numbers between a and b, including a $\left[a,b\right)$
${a}\lt{x}\le{ b}$ All real numbers between a and b, including b $\left(a,b\right]$
${a}\le{x}\le{ b}$ All real numbers between a and b, including a and b $\left[a,b\right]$
${x}\lt{a}\text{ or }{x}\gt{ b}$ All real numbers less than a or greater than b $\left(-\infty ,a\right)\cup \left(b,\infty \right)$
All real numbers All real numbers $\left(-\infty ,\infty \right)$

### Example: Using Interval Notation to Express an inequality

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.

### example: using interval notation to express an inequality

Describe the inequality $x\ge 4$ using interval notation

Answer: The solutions to $x\ge 4$ are represented as $\left[4,\infty \right)$. Note the use of a bracket on the left because 4 is included in the solution set.

### Try It

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

[ohm_question]58-92604[/ohm_question]

### Example: Using Interval Notation to Express a compound inequality

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.

### try it

We are going to look at a line with endpoints along the x-axis.
1. First we will adjust the left endpoint to (-15,0), and the right endpoint to (5,0)
2. Write an inequality that represents the line you created.
[practice-area rows="1"][/practice-area] 3. If we were to slide the left endpoint to (2,0), what do you think will happen to the line? [practice-area rows="1"][/practice-area] 4. Now what if we were to slide the right endpoint to (11,0), what do you think will happen to the line? Sketch on a piece of paper what you think this new inequality graph will look like. [practice-area rows="1"][/practice-area]

Answer: With endpoints (-15,0) and (5,0), the values for x on the line are between -15 and 5, so we can write $-15<x<5$. We made it a strict inequality because the dots on the endpoints of the lines are open. Moving the left endpoint towards the right endpoint shortens the line. Then moving the right endpoint away from the left endpoint lengthens the line again.

In the previous examples you were given an inequality or a description of one with words and asked to draw the corresponding graph and write the interval. In this example you are given an interval and asked to write the inequality and draw the graph. Given $\left(-\infty,10\right)$, write the associated inequality and draw the graph. In the box below, write down whether you think it will be easier to draw the graph first or write the inequality first. [practice-area rows="1"][/practice-area]

Answer: We will draw the graph first. The interval reads "all real numbers less than 10," so we will start by placing an open dot on 10 and drawing a line to the left with an arrow indicating the solution continues to negative infinity. To write the inequality, we will use < since the parentheses indicate that 10 is not included. $x<10$

In the following video, you will see examples of how to write inequalities in the three ways presented here: as an inequality, in interval notation, and with a graph. https://youtu.be/X0xrHKgbDT0

## Using the Properties of Inequalities

### recall solving multi-step equations

When solving inequalities, all the properties of equality and real numbers apply. We are permitted to add, subtract, multiply, or divide the same quantity to both sides of the inequality. Likewise, we may apply the distributive, commutative, and associative properties as desired to help isolate the variable. We may also distribute the LCD on both sides of an inequality to eliminate denominators. The only difference is that if we multiply or divide both sides by a negative quantity, we must reverse the direction of the inequality symbol.
When we work with inequalities, we can usually treat them similarly to but not exactly as we treat equations. 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, we must reverse the inequality symbol.

### A General Note: Properties of Inequalities

$\begin{array}{ll}\text{Addition Property}\hfill& \text{If }a< b,\text{ then }a+c< b+c.\hfill \\ \hfill & \hfill \\ \text{Multiplication Property}\hfill & \text{If }a< b\text{ and }c> 0,\text{ then }ac< bc.\hfill \\ \hfill & \text{If }a< b\text{ and }c< 0,\text{ then }ac> bc.\hfill \end{array}$

These properties also apply to $a\le b$, $a>b$, and $a\ge b$.

### Example: Demonstrating the Addition Property

Illustrate the addition property for inequalities by solving each of the following:
1. $x - 15<4$
2. $6\ge x - 1$
3. $x+7>9$

Answer: The addition property for inequalities states that if an inequality exists, adding or subtracting the same number on both sides does not change the inequality. 1. [latex-display]\begin{array}{ll}x - 15<4\hfill & \hfill \\ x - 15+15<4+15 \hfill & \text{Add 15 to both sides.}\hfill \\ x<19\hfill & \hfill \end{array}[/latex-display] 2. [latex-display]\begin{array}{ll}6\ge x - 1\hfill & \hfill \\ 6+1\ge x - 1+1\hfill & \text{Add 1 to both sides}.\hfill \\ 7\ge x\hfill & \hfill \end{array}[/latex-display] 3. [latex-display]\begin{array}{ll}x+7>9\hfill & \hfill \\ x+7 - 7>9 - 7\hfill & \text{Subtract 7 from both sides}.\hfill \\ x>2\hfill & \hfill \end{array}[/latex-display]

### Try It

Solve $3x - 2<1$.

[ohm_question]92605[/ohm_question]

### 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. $5-x>10$

Answer: 1. [latex-display]\begin{array}{l}3x<6\hfill \\ \frac{1}{3}\left(3x\right)<\left(6\right)\frac{1}{3}\hfill \\ x<2\hfill \end{array}[/latex-display] 2. [latex-display]\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}[/latex-display] 3. [latex-display]\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}[/latex-display]

### Try It

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

Answer: $x\ge -5$

[ohm_question]92606[/ohm_question]
Watch the following two videos for a demonstration of using the addition and multiplication properties to solve inequalities. [embed]https://youtu.be/1Z22Xh66VFM[/embed] [embed]https://youtu.be/RBonYKvTCLU[/embed]

## 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.

### Example: Solving an Inequality Algebraically

Solve the inequality: $13 - 7x\ge 10x - 4$.

Answer: Solving this inequality is similar to solving an equation up until the last step.

$\begin{array}{ll}13 - 7x\ge 10x - 4\hfill & \hfill \\ 13 - 17x\ge -4\hfill & \text{Move variable terms to one side of the inequality}.\hfill \\ -17x\ge -17\hfill & \text{Isolate the variable term}.\hfill \\ x\le 1\hfill & \text{Dividing both sides by }-17\text{ reverses the inequality}.\hfill \end{array}$
The solution set is given by the interval $\left(-\infty ,1\right]$, or all real numbers less than and including 1.

### Try It

Solve the inequality and write the answer using interval notation: $-x+4<\frac{1}{2}x+1$.

Answer: $\left(2,\infty \right)$

[ohm_question]92607[/ohm_question]

### 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: $-\frac{5}{6}x\le \frac{3}{4}+\frac{8}{3}x$.

[ohm_question]72891[/ohm_question]