Absolute Value Inequalities No Solution

The accented number of a number a is written as

$$\left | a \correct |$$

And represents the distance between a and 0 on a number line.

An absolute value equation is an equation that contains an absolute value expression. The equation

$$\left | x \right |=a$$

Has two solutions x = a and x = -a because both numbers are at the distance a from 0.

To solve an absolute value equation as

$$\left | x+7 \right |=14$$

You lot begin by making information technology into two split up equations then solving them separately.

$$x+7 =14$$

$$x+seven\, {\color{green} {-\, 7}}\, =14\, {\color{green} {-\, 7}}$$

$$x=7$$

or

$$x+7 =-14$$

$$10+7\, {\color{greenish} {-\, 7}}\, =-fourteen\, {\colour{green} {-\, 7}}$$

$$10=-21$$

An absolute value equation has no solution if the absolute value expression equals a negative number since an absolute value can never be negative.

The inequality

$$\left | x \right |<2$$

Represents the altitude betwixt x and 0 that is less than 2

Whereas the inequality

$$\left | 10 \right |>2$$

Represents the distance between x and 0 that is greater than 2

You can write an absolute value inequality as a compound inequality.

$$\left | 10 \right |<2\: or

$$-2<x<2$$

This holds true for all absolute value inequalities.

$$\left | ax+b \right |<c,\: where\: c>0$$

$$=-c<ax+b<c$$

$$\left | ax+b \correct |>c,\: where\: c>0$$

$$=ax+b<-c\: or\: ax+b>c$$

You lot tin can replace > above with ≥ and < with ≤.

When solving an absolute value inequality it'southward necessary to showtime isolate the accented value expression on one side of the inequality before solving the inequality.

---

Example

Solve the absolute value inequality

$$two\left |3x+9 \right |<36$$

$$\frac{2\left |3x+nine \correct |}{2}<\frac{36}{2}$$

$$\left | 3x+nine \right |<18$$

$$-18<3x+9<18$$

$$-18\, {\colour{green} {-\, 9}}<3x+9\, {\colour{green} {-\, 9}}<18\, {\colour{green} {-\, ix}}$$

$$-27<3x<9$$

$$\frac{-27}{{\colour{green} 3}}<\frac{3x}{{\colour{greenish} 3}}<\frac{9}{{\colour{green} 3}}$$

$$-nine<10<three$$

---

Video lesson

Solve the accented value equation

$$4 \left |2x -1 \right | -two = 10$$

Absolute Value Inequalities No Solution,

Source: https://www.mathplanet.com/education/algebra-1/linear-inequalities/solving-absolute-value-equations-and-inequalities

Posted by: nugentwhimsood.blogspot.com

Share this post