Solution for the question - solve and graph the equation : 3 > |x|-6

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Question

Solve and graph the equation : $3 greater than vertical line x vertical line minus 6$

hint Hint:

|x| is known as the absolute value of x. It is the non-negative value of x irrespective of its sign. The value of absolute value of x is given by
$vertical line x vertical line equals open curly brackets table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell negative x comma x less than 0 end cell row cell x comma x greater or equal than 0 end cell end table close$
First, we solve this inequality by considering the two cases. Then we plot the graph on the x- axis, or the real line R in such a way that the graph satisfies the value of x from both the cases.

The correct answer is: Combining the above two solutions, we get -9< x < 9

Step by step solution:
The given inequality is
$3 greater than vertical line x vertical line minus 6$
Adding 6 both sides, we have
$vertical line x vertical line less than 6 plus 3$
That is
$vertical line x vertical line less than 9$
We use the definition of $vertical line x vertical line$ , which is
$vertical line x vertical line equals open curly brackets table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell negative x comma x less than 0 end cell row cell x comma x greater or equal than 0 end cell end table close$
For, $x less than 0$ ,
We have
$vertical line x vertical line equals negative x less than 9$
Multiplying on both sides, we have
$x greater than negative 9$
Or
$negative 9 less than x$
For, $x greater or equal than 0$ ,
We have
$vertical line x vertical line equals x less than 9$
That is
$x less than 9$
Combining the above two solutions, we get
$negative 9 less than x less than 9$
We plot the above inequality on the real line.

The points -9 and 9 are not included in the graph.

The given inequality contains only one variable. So, the graph is plotted on one dimension, which is the real line. Geometrically, the absolute value of a number may be considered as its distance from zero regardless of its direction. The symbol |.| is pronounced as ‘modulus’. We read |x| as ‘modulus of x’ or ‘mod x’.

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Solve and graph the equation :

The correct answer is: Combining the above two solutions, we get -9< x < 9

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