Algebric questions on absolute value inequalities with solution
Answers
Answer:
The absolute number of a number a is written as
$$\left | a \right |$$
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 begin by making it into two separate equations and then solving them separately.
$$x+7 =14$$
$$x+7\, {\color{green} {-\, 7}}\, =14\, {\color{green} {-\, 7}}$$
$$x=7$$
or
$$x+7 =-14$$
$$x+7\, {\color{green} {-\, 7}}\, =-14\, {\color{green} {-\, 7}}$$
$$x=-21$$
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 distance between x and 0 that is less than 2
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Whereas the inequality
$$\left | x \right |>2$$
Represents the distance between x and 0 that is greater than 2
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You can write an absolute value inequality as a compound inequality.
$$\left | x \right |<2\: or
$$-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 \right |>c,\: where\: c>0$$
$$=ax+b<-c\: or\: ax+b>c$$
You can replace > above with ≥ and < with ≤.
When solving an absolute value inequality it's necessary to first isolate the absolute value expression on one side of the inequality before solving the inequality.
example-
Solve the absolute value inequality
Solve the absolute value inequality$$2\left |3x+9 \right |<36$$
Solve the absolute value inequality$$2\left |3x+9 \right |<36$$$$\frac{2\left |3x+9 \right |}{2}<\frac{36}{2}$$
Solve the absolute value inequality$$2\left |3x+9 \right |<36$$$$\frac{2\left |3x+9 \right |}{2}<\frac{36}{2}$$$$\left | 3x+9 \right |<18$$
Solve the absolute value inequality$$2\left |3x+9 \right |<36$$$$\frac{2\left |3x+9 \right |}{2}<\frac{36}{2}$$$$\left | 3x+9 \right |<18$$$$-18<3x+9<18$$
Solve the absolute value inequality$$2\left |3x+9 \right |<36$$$$\frac{2\left |3x+9 \right |}{2}<\frac{36}{2}$$$$\left | 3x+9 \right |<18$$$$-18<3x+9<18$$$$-18\, {\color{green} {-\, 9}}<3x+9\, {\color{green} {-\, 9}}<18\, {\color{green} {-\, 9}}$$
Solve the absolute value inequality$$2\left |3x+9 \right |<36$$$$\frac{2\left |3x+9 \right |}{2}<\frac{36}{2}$$$$\left | 3x+9 \right |<18$$$$-18<3x+9<18$$$$-18\, {\color{green} {-\, 9}}<3x+9\, {\color{green} {-\, 9}}<18\, {\color{green} {-\, 9}}$$$$-27<3x<9$$
Solve the absolute value inequality$$2\left |3x+9 \right |<36$$$$\frac{2\left |3x+9 \right |}{2}<\frac{36}{2}$$$$\left | 3x+9 \right |<18$$$$-18<3x+9<18$$$$-18\, {\color{green} {-\, 9}}<3x+9\, {\color{green} {-\, 9}}<18\, {\color{green} {-\, 9}}$$$$-27<3x<9$$$$\frac{-27}{{\color{green} 3}}<\frac{3x}{{\color{green} 3}}<\frac{9}{{\color{green} 3}}$$
Solve the absolute value inequality$$2\left |3x+9 \right |<36$$$$\frac{2\left |3x+9 \right |}{2}<\frac{36}{2}$$$$\left | 3x+9 \right |<18$$$$-18<3x+9<18$$$$-18\, {\color{green} {-\, 9}}<3x+9\, {\color{green} {-\, 9}}<18\, {\color{green} {-\, 9}}$$$$-27<3x<9$$$$\frac{-27}{{\color{green} 3}}<\frac{3x}{{\color{green} 3}}<\frac{9}{{\color{green} 3}}$$$$-9<x<3$$