Coordinate Algebra Absolute Value Equations and Inequalities Review

Learning Objectives

In this section, yous volition:

• Utilize interval notation
• Use properties of inequalities.
• Solve inequalities in ane variable algebraically.
• Solve absolute value inequalities.

Figure one

It is not easy to make the honor gyre at most top universities. Suppose students were required to comport a course load of at least 12 credit hours and maintain a grade point average of 3.5 or higher up. How could these award whorl requirements exist 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 $\mathrm{10}\ge 4$ tin can be achieved in several ways.

We can apply a number line as shown in Figure 2. The bluish ray begins at $\mathrm{ten}=4$ and, as indicated by the arrowhead, continues to infinity, which illustrates that the solution set includes all existent numbers greater than or equal to 4.

Figure 2

We can use prepare-builder note: $\left\{x|\mathrm{10}\ge 4\right\},$ which translates to "all real numbers x such that 10 is greater than or equal to four." Notice that braces are used to indicate a set.

The tertiary method is interval notation, in which solution sets are indicated with parentheses or brackets. The solutions to $x\ge 4$ are represented every bit $\left[4,\infty \right).$ This is perhaps the most useful method, every bit it applies to concepts studied later in this grade and to other higher-level math courses.

The primary concept to call up is that parentheses stand for solutions greater or less than the number, and brackets stand for 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 exist "equaled." A few examples of an interval, or a set of numbers in which a solution falls, are $\left[\mathrm{-2},\mathrm{half-dozen}\right),$ or all numbers between $\mathrm{-2}$ and $6,$ including $\mathrm{-ii},$ but not including $\mathrm{vi};$ $\left(-\mathrm{ane},0\right),$ all real numbers between, only not including $\mathrm{-1}$ and $0;$ and $\left(-\infty ,\mathrm{ane}\right],$ all real numbers less than and including $1.$ Table i outlines the possibilities.

Set Indicated	Set-Architect Notation	Interval Annotation
All existent numbers between a and b, only not including a or b	$\left\{\mathrm{ten}|a<\mathrm{10}<b\right\}$	$\left(a,b\right)$
All real numbers greater than a, merely non including a	$\left\{x|x>a\right\}$	$\left(a,\infty \right)$
All real numbers less than b, simply not including b	$\left\{x|x<b\right\}$	$\left(-\infty ,b\right)$
All real numbers greater than a, including a	$\left\{x|x\ge a\right\}$	$\left[a,\infty \right)$
All existent numbers less than b, including b	$\left\{x|x\le b\right\}$	$\left(-\infty ,b\right]$
All real numbers between a and b, including a	$\left\{x|a\le x<b\right\}$	$\left[a,b\right)$
All real numbers between a and b, including b	$\left\{x|a<x\le b\right\}$	$\left(a,b\right]$
All real numbers between a and b, including a and b	$\left\{x|a\le x\le b\right\}$	$\left[a,b\right]$
All real numbers less than a or greater than b	$\left\{\mathrm{ten}|x<a\text{or}x>b\right\}$	$\left(-\infty ,a\right)\cup \left(b,\infty \right)$
All real numbers	$\left\{\mathrm{ten}|\mathrm{10}\text{is all real numbers}\right\}$	$\left(-\infty ,\infty \right)$

Table 1

Example 1

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

Use interval note to indicate all real numbers greater than or equal to $\mathrm{-2.}$

Try Information technology #1

Use interval notation to indicate all existent numbers betwixt and including $\mathrm{-three}$ and $5.$

Example 2

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 existent numbers less than or equal to $\mathrm{-i}$ or greater than or equal to $1.$

Try It #2

Express all real numbers less than $\mathrm{-2}$ or greater than or equal to iii in interval notation.

Using the Properties of Inequalities

When nosotros work with inequalities, we can usually treat them similarly to only not exactly as we treat equalities. Nosotros can utilise the addition holding and the multiplication property to help u.s. solve them. The 1 exception is when we multiply or split by a negative number; doing then reverses the inequality symbol.

Backdrop of Inequalities

$$\begin{array}{ll}Additio\mathrm{due\; north}Property\hfill & \text{If}a<b,\text{then}a+c<b+c.\hfill \\ \hfill & \hfill \\ \mathrm{One\; thousand}ultiplicationProperty\hfill & \text{If}a<b\text{and}c>0,\text{so}ac<bc.\hfill \\ \hfill & \text{If}a<b\text{and}c<0,\text{and so}ac>bc.\hfill \end{array}$$

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

Example 3

Demonstrating the Addition Property

Illustrate the add-on holding for inequalities past solving each of the following:

1. ⓐ $x-15<4$
2. ⓑ $6\ge \mathrm{ten}-1$
3. ⓒ $x+\mathrm{vii}>9$

Try It #3

Example 4

Demonstrating the Multiplication Property

Illustrate the multiplication property for inequalities by solving each of the following:

1. ⓐ $3x<6$
2. ⓑ $\mathrm{-two}x-1\ge \mathrm{five}$
3. ⓒ $\mathrm{v}-x>10$

Try It #4

Solve: $4x+7\ge 2x-\mathrm{iii.}$

Solving Inequalities in One Variable Algebraically

Every bit the examples accept shown, we can perform the same operations on both sides of an inequality, just as nosotros practice with equations; nosotros combine like terms and perform operations. To solve, we isolate the variable.

Example 5

Solving an Inequality Algebraically

Solve the inequality: $13-\mathrm{seven}x\ge 10x-\mathrm{iv.}$

Try It #5

Solve the inequality and write the respond using interval annotation: $-x+4<\frac{1}{2}\mathrm{ten}+1.$

Example 6

Solving an Inequality with Fractions

Solve the post-obit inequality and write the answer in interval notation: $-\frac{3}{4}\mathrm{ten}\ge -\frac{5}{8}+\frac{\mathrm{ii}}{3}\mathrm{10}.$

Try It #half-dozen

Solve the inequality and write the respond in interval annotation: $-\frac{\mathrm{v}}{6}\mathrm{ten}\le \frac{\mathrm{iii}}{\mathrm{iv}}+\frac{\mathrm{eight}}{3}x.$

Understanding Chemical compound Inequalities

A chemical compound inequality includes ii inequalities in one statement. A argument such as $\mathrm{iv}<\mathrm{ten}\le \mathrm{six}$ means $\mathrm{iv}<\mathrm{10}$ and $x\le \mathrm{half-dozen.}$ In that location are two ways to solve chemical 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.

Example 7

Solving a Compound Inequality

Solve the compound inequality: $3\le 2x+2<6.$

Effort It #vii

Solve the chemical compound inequality: $4<2x-8\le \mathrm{x.}$

Example 8

Solving a Compound Inequality with the Variable in All Three Parts

Solve the compound inequality with variables in all three parts: $\mathrm{three}+\mathrm{ten}>\mathrm{vii}x-\mathrm{two}>5\mathrm{ten}-\mathrm{x.}$

Effort It #eight

Solve the chemical compound inequality: $3y<\mathrm{iv}-\mathrm{five}y<5+3y.$

Solving Accented Value Inequalities

Equally we know, the absolute value of a quantity is a positive number or null. From the origin, a betoken located at $\left(-x,0\right)$ has an absolute value of $x,$ as information technology is x units away. Consider accented value as the distance from i point to some other point. Regardless of direction, positive or negative, the altitude betwixt the ii points is represented as a positive number or nil.

An absolute value inequality is an equation of the form

$$\left|A\right|<B,\left|A\right|\le B,\left|A\right|>B,\text{or}\left|A\right|\ge B,$$

Where A, and sometimes B, represents an algebraic expression dependent on a variable x. Solving the inequality means finding the fix of all $\mathrm{ten}$ -values that satisfy the problem. Usually this set will exist an interval or the union of ii intervals and volition include a range of values.

At that place are two bones 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 arroyo is that solutions are exact, as precise solutions are sometimes hard to read from a graph.

Suppose we want to know all possible returns on an investment if nosotros could earn some corporeality of money inside $200 of $600. Nosotros can solve algebraically for the ready of x-values such that the distance between $x$ and 600 is less than or equal to 200. We represent the distance between $x$ and 600 as $\left|x-600\right|,$ and therefore, $\left|x-600\right|\le 200$ or

$$\begin{array}{c}\mathrm{-200}\le x-600\le 200\\ \mathrm{-200}+600\le x-600+600\le 200+600\\ 400\le x\le 800\end{array}$$

This ways our returns would be betwixt $400 and $800.

To solve absolute value inequalities, simply as with absolute value equations, nosotros write two inequalities and so solve them independently.

Absolute Value Inequalities

For an algebraic expression X, and $\mathrm{yard}>0,$ an absolute value inequality is an inequality of the class

$$\begin{array}{l}\left|X\right|<\mathrm{one\; thousand}\text{is equivalent to}-\mathrm{grand}<X<\mathrm{chiliad}\hfill \\ \left|X\right|>k\text{is equivalent to}X<-g\text{or}X>k\hfill \end{array}$$

These statements likewise utilise to $\left|\mathrm{10}\right|\le \mathrm{yard}$ and $\left|X\right|\ge k.$

Case 9

Determining a Number within a Prescribed Altitude

Describe all values $\mathrm{ten}$ within a distance of four from the number 5.

Try It #ix

Describe all x-values inside a distance of 3 from the number ii.

Example 10

Solving an Absolute Value Inequality

Solve $|\mathrm{ten}-\mathrm{i}|\le 3$ .

Example eleven

Using a Graphical Arroyo to Solve Absolute Value Inequalities

Given the equation $y=-\frac{1}{2}|\mathrm{iv}x-5|+3,$ determine the x-values for which the y-values are negative.

Endeavour Information technology #ten

Solve $-\mathrm{ii}\left|m-4\right|\le -6.$

2.7 Section Exercises

Verbal

1 .

When solving an inequality, explain what happened from Stride 1 to Step 2:

$\begin{array}{ll}\text{Step 1}\hfill & -\mathrm{ii}x>6\hfill \\ \text{Step 2}\hfill & x<-3\hfill \end{array}$

2 .

When solving an inequality, we arrive at:

$\begin{array}{l}x+\mathrm{two}<\mathrm{10}+3\hfill \\ \mathrm{ii}<3\hfill \end{array}$

Explain what our solution ready is.

3 .

When writing our solution in interval notation, how do nosotros represent all the real numbers?

4 .

When solving an inequality, we arrive at:

$\begin{array}{l}x+2>x+\mathrm{iii}\hfill \\ 2>3\hfill \end{array}$

Explicate what our solution prepare is.

5 .

Describe how to graph $y=\left|x-3\right|$

Algebraic

For the following exercises, solve the inequality. Write your final answer in interval notation.

9 .

$\mathrm{four}(\mathrm{10}+\mathrm{iii})\ge 2\mathrm{10}-1$

x .

$-\frac{1}{2}\mathrm{ten}\le -\frac{5}{4}+\frac{\mathrm{ii}}{\mathrm{v}}\mathrm{ten}$

11 .

$\mathrm{-v}(x-1)+\mathrm{iii}>\mathrm{three}x-4-4\mathrm{10}$

12 .

$\mathrm{-3}(\mathrm{two}x+\mathrm{ane})>\mathrm{-2}(x+4)$

xiii .

$\frac{x+3}{8}-\frac{x+\mathrm{five}}{5}\ge \frac{\mathrm{iii}}{10}$

fourteen .

$\frac{x-\mathrm{i}}{3}+\frac{\mathrm{ten}+2}{5}\le \frac{\mathrm{iii}}{\mathrm{v}}$

For the following exercises, solve the inequality involving absolute value. Write your final answer in interval notation.

15 .

$\left|x+9\right|\ge \mathrm{-half\; dozen}$

xvi .

$\left|2x+3\right|<\mathrm{vii}$

18 .

$\left|\mathrm{ii}x+1\right|+\mathrm{i}\le 6$

19 .

$\left|x-\mathrm{ii}\right|+\mathrm{four}\ge 10$

20 .

$\left|\mathrm{-ii}x+7\right|\le 13$

23 .

$\left|\frac{x-3}{4}\right|<2$

For the following exercises, draw all the ten-values within or including a distance of the given values.

24 .

Altitude of v units from the number vii

25 .

Altitude of 3 units from the number ix

26 .

Altitude of 10 units from the number 4

27 .

Distance of xi units from the number 1

For the following exercises, solve the chemical compound inequality. Limited your answer using inequality signs, and so write your respond using interval annotation.

28 .

$\mathrm{-4}<3x+2\le 18$

29 .

$\mathrm{three}\mathrm{ten}+1>\mathrm{ii}x-5>x-7$

30 .

$3y<5-2y<7+y$

31 .

$2x-5<\mathrm{-11}\text{or}\mathrm{five}\mathrm{ten}+1\ge 6$

Graphical

For the following exercises, graph the function. Observe the points of intersection and shade the x-axis representing the solution fix to the inequality. Show your graph and write your final answer in interval notation.

For the following exercises, graph both direct lines (left-hand side being y1 and correct-hand side being y2) on the same axes. Detect the point of intersection and solve the inequality by observing where it is true comparing the y-values of the lines.

41 .

$\frac{\mathrm{one}}{2}x+1>\frac{1}{\mathrm{two}}x-5$

42 .

$4x+1<\frac{1}{2}\mathrm{ten}+3$

Numeric

For the following exercises, write the set in interval notation.

43 .

$\{x|\mathrm{-ane}<\mathrm{10}<\mathrm{iii}\}$

44 .

$\{x|\mathrm{10}\ge \mathrm{vii}\}$

45 .

$\{x|\mathrm{ten}<4\}$

46 .

$\{x|\mathrm{10}\text{is all real numbers}\}$

For the following exercises, write the interval in set-builder notation.

50 .

$[\mathrm{-4},\mathrm{one}]\cup [9,\infty )$

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

52 .

Technology

For the post-obit exercises, input the left-hand side of the inequality as a Y1 graph in your graphing utility. Enter y2 = the correct-hand side. Entering the absolute value of an expression is institute in the MATH menu, Num, 1:abs(. Find the points of intersection, retrieve (2^nd CALC 5:intersection, one^st curve, enter, ii^nd curve, enter, judge, enter). Re-create a sketch of the graph and shade the x-axis for your solution set to the inequality. Write terminal answers in interval notation.

54 .

$\left|\mathrm{10}+2\right|-\mathrm{v}<2$

55 .

$\frac{-1}{\mathrm{ii}}\left|x+\mathrm{two}\right|<\mathrm{four}$

56 .

$\left|\mathrm{four}x+\mathrm{one}\right|-3>\mathrm{ii}$

Extensions

59 .

Solve $\left|\mathrm{three}x+1\right|=\left|2x+3\right|$

60 .

Solve $x^{\mathrm{ii}}-x>12$

61 .

$\frac{x-5}{x+7}\le 0,$ $x\ne \mathrm{-seven}$

62 .

$p=-x^2+130x-3000$ is a profit formula for a minor concern. Find the set of x-values that will keep this turn a profit positive.

Real-Globe Applications

63 .

In chemistry the volume for a certain gas is given by $V=20T,$ where V is measured in cc and T is temperature in ºC. If the temperature varies between 80ºC and 120ºC, notice the set of volume values.

64 .

A basic cellular parcel costs $20/mo. for 60 min of calling, with an boosted accuse of $.thirty/min beyond that fourth dimension.. The toll formula would be $$C=\mathrm{twenty}+.30(\mathrm{ten}-\mathrm{threescore}).$$ If y'all have to keep your bill no greater than $50, what is the maximum calling minutes you tin use?

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Source: https://openstax.org/books/college-algebra/pages/2-7-linear-inequalities-and-absolute-value-inequalities

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