Answer:
A) -5[tex]\leq[/tex] x < 1
Step-by-step explanation:
First, let's notice that we need positives numbers inside both roots. The root of a negative number is a math error (we are not dealing with complex numbers right now). With that information, let's analyze the options:
A) -5[tex]\leq[/tex] x < 1
If we add 5 in this inequation, we have
-5+ 5[tex]\leq[/tex] x+5 < 1+5
0[tex]\leq[/tex] x+5 < 6
That's means the number in the first root is positive, great!
Now, we want 1-x to be positive:
-5[tex]\leq[/tex] x < 1
5[tex]\geq[/tex] -x > -1 (remember that the inequation changes side when we multiply by a negative number, (-1) in this case)
1+5[tex]\geq[/tex] 1-x > 1-1
6[tex]\geq[/tex] 1-x > 0
It's positive!
This is the correct answer but let's see why the others are incorrect.
B)5> x [tex]\leq[/tex]-1
We just need to find a number that fits the inequation but causes at least one root to have a negative number inside.
For example, -6.
5>-6
-6 [tex]\leq[/tex] -1
But -6+5 = -1 is negative
C) 5 > x > 1
In this case, the number 2 fits the inequation but it causes the second root to have a negative number inside, 1 - 2 = -1
D) 5 [tex]\leq[/tex] x [tex]\leq[/tex] 1
This option can't be correct because there is no number smaller than 1 and bigger than 5.
