solve the inequality.then graph the solution set |x-8|-8 less then equal to -3

Question

solve the inequality. then graph the solution set |x-8|-8 less then equal to -3

Answer 1

The inequality, |x-8|-8≤-3, can be solved as follows:

 

|x-8|≤-3 +8          add 8 to both sides

|x - 8|≤ 5             Now since |a|≤1 can be simplified as -1 ≤ a ≤ 1, let us apply this to the inequality

 

-5 ≤ x - 8 ≤ 5        Drop the absolute value and rewrite the inequality

3  ≤  x  ≤  13         add 8 to all the sides, including the middle expression

 

In other words, this inequality is true for all real numbers between 3 and 13, including 3 and 13.

The question requires a graph, therefore, to graph the solution set draw a NUMBER LINE.

1. On the number line mark the numbers 3 and 13, left to right,
2. Mark a SOLID DOT on 3 and 13, then
3. Shade the section of the number line between 3 and 13

 

Remember, the question is giving you x as a variable; it could as easily been other variables- a, b, c

 

Note: if the question had been presented as a linear inequality, for instance y ≤ | x - 8|, you would graph the related linear equation on a coordinate plane and shade the region where the solution points are.

Answer 2

Note that /a/<n   means that    -n< a < n

 

/x-8/-8<-3

 

First add 8 to both sides to get the absolute value alone

 

/x-8/ < -3+8

/x-8/<5

 

This means that x-8 < 5    or    x-8 > -5

Together this would  -5 < x-8 < 5

 

Take this as 2 inequalities.... and solve them separately

 

x-8<5        x-8>-5

x<5+8       x>-5+8

x<13         x>3

 

Together this is   3<x<13

 

To graph this you would draw a dashed vertical line at x=3

and another dashed vertical line at x = 13

The solution set is all x greater than 3 and less than 13 so shade

the area between the dashed lines.

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Note:  The lines are dashed instead of solid to indicate that x≠3 and x≠13

x can only be greater than 3 and less than 13 but not equal to either.

 

If our inequality had been /x-8/-8 ≤ -3 then our final answer would

have been     3 ≤ x ≤ 13 and the vertical lines would be drawn solid

to indicate that x can be equal to the end values.