can you help me with math please.

Question

(a) The solution of the inequality  |x| ≤ 7 is the following interval. answer using( INTERVAL NOTATION )


(b) The solution of the inequality  |x| ≥ 7 is a union of two intervals. (State the solution. the answer using INTERVAL NOTATION.)

Answer 1

(a)  All |x|≤7 is saying is that the absolute value of x is less than or equal to 7.   For absolute value, it is often helpful to consider two cases: (1) x ≥ 0 and (2) x < 0 

 

For x≥0,  |x| = x,  and  0 ≤ x ≤ 7 

For x<0,  |x| = -x,  and  -7 ≤ x < 0

 

Thus for  |x| ≤ 7,  -7 ≤ x ≤ 7.   In interval notation, x lies in the interval [-7, 7]

 

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(b) Yes, this is the union of two regions.   If you think of a number line, the solution to this part is very close to being the inverse of part (a)  (with the exception of the two points x=-7 and x=7).   Think about it for a while and message me if you are stuck.  

 

 

Answer 2

When dealing with absolute value in inequalities, we have to set what's inside to negative, also, to account that two opposite values share the same absolute value.

 

In this case, it's simple.

 

x ≥ 7

 

-x ≥ 7

x ≤ 7 (dividing by a negative reverses the sign)

 

(-∞,-7] U [7,∞)