## 17Calculus Precalculus - Absolute Value

Absolute values in functions are not hard to handle. You just need to get used to the notation and learn how to break up a function into different parts.

What Does 'Absolute Value' Mean?

The absolute value signs are two vertical lines on both sides of an expression, for example $$\abs{x+3}$$. When an expression is contained in a set of absolute value signs the idea is that the sign of the expression is stripped off and replaced with a positive sign. There are two possible cases.
Case 1 If the expression is already positive or zero, the absolute value signs do nothing and can be dropped.
Case 2 If the expression is less than zero, then the expression is replaced by the negative of itself and the absolute value signs can then be dropped.

Example 1: $$\abs{3} = 3$$
Since $$3$$ is already positive, the absolute value signs do nothing and therefore can be dropped with no change in the expression, in this case $$3$$.
Example 2: $$\abs{-3} = 3$$
In this case, the expression $$-3$$ is stripped of the negative sign and replaced with the positive version of the expression, in this case the new value is $$3$$.

These examples are pretty easy to see what needs to be done with the sign of the expression but most times, we do not know if the expression is positive or negative.
Example 3: $$\abs{x}$$
Since x is a variable and it's value has not been limited as positive, negative or even real, we can do nothing about the absolute value signs. The absolute value signs are a shorthand way of saying that when $$x$$ is positive, the expression is just $$x$$ but when $$x$$ is negative, the expression is $$-x$$.

This video clip explains this idea in more detail very well and shows a great way to write the absolute value function as a piecewise function.

### Dr Chris Tisdell - What is the absolute value function? (part 1) [1min-57secs]

video by Dr Chris Tisdell

Notice he said in the video that the absolute value tells us the distance between things. This is a great way of looking at it.

Graphs of Absolute Value Functions

Let's watch a bit more of the same video above discussing how to graph absolute value functions and how to visualize the distance idea on the graph.

### Dr Chris Tisdell - What is the absolute value function? (part 2) [about 9min]

video by Dr Chris Tisdell

Solving Absolute Value Equations

### MIP4U - Absolute Value Equations [9min-30secs]

video by MIP4U

After that video, you should be ready for some practice problems.

Practice Solving Equalities

Basic

Unless otherwise instructed, find all values of $$x$$ that solve each equation or inequality.

$$\abs{x-2}=4$$

Problem Statement

Find the values of $$x$$ that solve $$|x-2|=4$$

$$x=-2, ~ x=6$$

Problem Statement

Find the values of $$x$$ that solve $$|x-2|=4$$

Solution

### 2087 video

video by Dr Chris Tisdell

$$x=-2, ~ x=6$$

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$$|3x-2|=4$$

Problem Statement

Find the values of $$x$$ that solve $$|3x-2|=4$$.

$$x=-2/3, ~ x=2$$

Problem Statement

Find the values of $$x$$ that solve $$|3x-2|=4$$.

Solution

### 2088 video

video by Dr Chris Tisdell

$$x=-2/3, ~ x=2$$

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$$|3x+2| = 8$$

Problem Statement

Find the values of $$x$$ that solve the equation $$|3x+2| = 8$$.

Solution

### 2093 video

video by PatrickJMT

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$$|7-2x| = 10$$

Problem Statement

Find the values of $$x$$ that solve the equation $$|7-2x| = 10$$.

$$x=-3/2, x=17/2$$

Problem Statement

Find the values of $$x$$ that solve the equation $$|7-2x| = 10$$.

Solution

### 2094 video

video by PatrickJMT

$$x=-3/2, x=17/2$$

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$$|3x-2| + 4 = 4$$

Problem Statement

Find the values of $$x$$ that solve the equation $$|3x-2| + 4 = 4$$.

Solution

### 2560 video

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$$|x+1| + 6 = 2$$

Problem Statement

Find the values of $$x$$ that solve the equation $$|x+1| + 6 = 2$$.

Solution

### 2561 video

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$$3|2x-1| = 21$$

Problem Statement

Find the values of $$x$$ that solve the equation $$3|2x-1| = 21$$.

Solution

### 2562 video

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$$|3x| = 18$$

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$|3x| = 18$$.

$$x = 6$$ and $$x = -6$$

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$|3x| = 18$$.

Solution

### 2886 video

$$x = 6$$ and $$x = -6$$

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$$|5x + 7| = 42$$

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$|5x + 7| = 42$$.

$$x = 7$$ and $$x = -49/5$$

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$|5x + 7| = 42$$.

Solution

### 2887 video

$$x = 7$$ and $$x = -49/5$$

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$$|2x - 3 | = 9$$

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$|2x - 3 | = 9$$.

$$x = 6$$ and $$x = -3$$

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$|2x - 3 | = 9$$.

Solution

### 2888 video

$$x = 6$$ and $$x = -3$$

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$$\displaystyle{ \left| \frac{4x+6}{5} \right| = 6}$$

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$\displaystyle{ \left| \frac{4x+6}{5} \right| = 6}$$.

$$x = 6$$ and $$x = -9$$

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$\displaystyle{ \left| \frac{4x+6}{5} \right| = 6}$$.

Solution

### 2889 video

$$x = 6$$ and $$x = -9$$

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$$2|3x - 4| + 5 = 27$$

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$2|3x - 4| + 5 = 27$$.

$$x = 5$$ and $$x = -7/3$$

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$2|3x - 4| + 5 = 27$$.

Solution

### 2890 video

$$x = 5$$ and $$x = -7/3$$

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$$3|4x - 1| - 5 = 16$$

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$3|4x - 1| - 5 = 16$$.

$$x = 2$$ and $$x = -3/2$$

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$3|4x - 1| - 5 = 16$$.

Solution

### 2891 video

$$x = 2$$ and $$x = -3/2$$

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$$|x-4| + 8 = 3$$

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$|x-4| + 8 = 3$$

no solution

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$|x-4| + 8 = 3$$

Solution

### 2892 video

no solution

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$$|7x + 2| = 4x + 11$$

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$|7x + 2| = 4x + 11$$.

$$x = 3$$ and $$x = -13/11$$

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$|7x + 2| = 4x + 11$$.

Solution

### 2893 video

$$x = 3$$ and $$x = -13/11$$

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$$|x + 3| = |x - 11|$$

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$|x + 3| = |x - 11|$$.

$$x = 4$$

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$|x + 3| = |x - 11|$$.

Solution

### 2894 video

$$x = 4$$

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Intermediate

$$|2x-1| = 3|4-8x| - 12$$

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$|2x-1| = 3|4-8x| - 12$$ giving your answer in exact form.

Solution

### 3884 video

video by Michael Penn

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$$|3x+6| + |x-1| = 10$$

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$|3x+6| + |x-1| = 10$$ giving your answer in exact form.

Solution

### 3885 video

video by Michael Penn

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Solving Absolute Value Inequalities

This guy does a great job of explaining how to solve inequalities containing absolute values.

### MIP4U - Absolute Value Inequalities [10min-29secs]

video by MIP4U

Now try these practice problems.

Practice Solving Inequalities

Unless otherwise instructed, find all values of $$x$$ that solve each equation or inequality.

$$|x-2| < 4$$

Problem Statement

Find the values of $$x$$ that solve the inequality $$|x-2| < 4$$.

$$-2 < x < 6$$

Problem Statement

Find the values of $$x$$ that solve the inequality $$|x-2| < 4$$.

Solution

### 2089 video

video by Dr Chris Tisdell

$$-2 < x < 6$$

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$$|3x-2| < 4$$

Problem Statement

Find the values of $$x$$ that solve the inequality $$|3x-2| < 4$$.

$$-2/3 < x < 2$$

Problem Statement

Find the values of $$x$$ that solve the inequality $$|3x-2| < 4$$.

Solution

### 2090 video

video by Dr Chris Tisdell

$$-2/3 < x < 2$$

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$$|1-3x| \leq 2$$

Problem Statement

Find the values of $$x$$ that solve the inequality $$|1-3x| \leq 2$$.

$$-1/3 \leq x \leq 1$$

Problem Statement

Find the values of $$x$$ that solve the inequality $$|1-3x| \leq 2$$.

Solution

### 2091 video

video by Dr Chris Tisdell

$$-1/3 \leq x \leq 1$$

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$$|5-4x| > 2$$

Problem Statement

Find the values of $$x$$ that solve the inequality $$|5-4x| > 2$$

$$x < 3/4$$ or $$x > 7/4$$

Problem Statement

Find the values of $$x$$ that solve the inequality $$|5-4x| > 2$$

Solution

### 2092 video

video by Dr Chris Tisdell

$$x < 3/4$$ or $$x > 7/4$$

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$$3|x+7| > 27$$

Problem Statement

Find the values of $$x$$ that solve the inequality $$3|x+7| > 27$$.

Solution

### 2558 video

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$$|x+3| \leq 4$$

Problem Statement

Find the values of $$x$$ that solve the inequality $$|x+3| \leq 4$$.

Solution

### 2559 video

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$$|x| < 4$$

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$|x| < 4$$.

$$-4 < x < 4$$

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$|x| < 4$$.

Solution

### 2895 video

$$-4 < x < 4$$

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$$|2x - 3| \geq 8$$

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$|2x - 3| \geq 8$$.

$$x \geq 5.5$$ or $$x \leq -2.5$$

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$|2x - 3| \geq 8$$.

Solution

### 2896 video

$$x \geq 5.5$$ or $$x \leq -2.5$$

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$$5 - 3|4x+1| \geq -9$$

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$5 - 3|4x+1| \geq -9$$.

$$-17/12 \leq x \leq 11/12$$

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$5 - 3|4x+1| \geq -9$$.

Solution

### 2897 video

$$-17/12 \leq x \leq 11/12$$

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$$|2x + 5| > 11$$

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$|2x + 5| > 11$$

$$x > 3$$ or $$x < -8$$

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$|2x + 5| > 11$$

Solution

### 2898 video

$$x > 3$$ or $$x < -8$$

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$$|3x - 4| \leq 17$$

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$|3x - 4| \leq 17$$.

$$13/3 \leq x \leq 7$$

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$|3x - 4| \leq 17$$.

Solution

### 2899 video

$$13/3 \leq x \leq 7$$

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$$|3x + 5| \leq -3$$

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$|3x + 5| \leq -3$$

no solution

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$|3x + 5| \leq -3$$

Solution

### 2900 video

no solution

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$$|4x - 3| > -4$$

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$|4x - 3| > -4$$

all real numbers

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$|4x - 3| > -4$$

Solution

### 2901 video

all real numbers

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$$3|7x - 4| +5 > 17$$

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$3|7x - 4| +5 > 17$$

$$x > 8/7$$ or $$x < 0$$

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$3|7x - 4| +5 > 17$$

Solution

### 2902 video

$$x > 8/7$$ or $$x < 0$$

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$$2 \leq |3x+5| \leq 10$$

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$2 \leq |3x+5| \leq 10$$ giving your answer in interval notation in exact form.

Solution

### 3862 video

video by Michael Penn

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$$||3x-1|+4| \lt 8$$

Problem Statement

Unless otherwise instructed, find the values of $$x$$ that solve $$||3x-1|+4| \lt 8$$ giving your answer in interval notation in exact form.

Solution

### 3863 video

video by Michael Penn

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Finally, if you want a full lecture on absolute value equations, inequalities and other inequalities, here is a video.

### freeCodeCamp.org - [39mins-18secs]

video by freeCodeCamp.org

Really UNDERSTAND Precalculus

 piecewise functions

### Trig Formulas

The Unit Circle

The Unit Circle [wikipedia] Basic Trig Identities

Set 1 - basic identities

$$\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }$$

$$\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }$$

$$\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }$$

$$\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }$$

Set 2 - squared identities

$$\sin^2t + \cos^2t = 1$$

$$1 + \tan^2t = \sec^2t$$

$$1 + \cot^2t = \csc^2t$$

Set 3 - double-angle formulas

$$\sin(2t) = 2\sin(t)\cos(t)$$

$$\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }$$

Set 4 - half-angle formulas

$$\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }$$

$$\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }$$

Trig Derivatives

 $$\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }$$ $$\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }$$ $$\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }$$ $$\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }$$ $$\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }$$ $$\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }$$

Inverse Trig Derivatives

 $$\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }$$ $$\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }$$

Trig Integrals

 $$\int{\sin(x)~dx} = -\cos(x)+C$$ $$\int{\cos(x)~dx} = \sin(x)+C$$ $$\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C$$ $$\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C$$ $$\int{\sec(x)~dx} =$$ $$\ln\abs{\sec(x)+\tan(x)}+C$$ $$\int{\csc(x)~dx} =$$ $$-\ln\abs{\csc(x)+\cot(x)}+C$$

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