Challenge 7: What values of x satisfy |x-4|+|x+4|<= 10.
Please give your response in intveral notation.
|x-4| ≤ 10 - |x+4|
x-4 ≤ 10-|x+4| AND -x+4 ≤ 10 - |x+4)
|x+4 ≤ 14-x AND |x+4| ≤ 6 + x
x+4≤14-x AND -x-4 ≤ 14-x AND x+4 ≤ 6+x AND -x-4 ≤ 6+x
2x ≤ 10 AND -4 ≤ 14 AND 4 ≤ 6 AND -2x ≤ 10
x ≤ 5 AND x ≥ -5
The two inner statements are true, so we just use the outer two.
I don't use this "new" interval notation, I prefer
-5 ≤ x ≤ 5
I will let you change it to your notation.
To find the values of x that satisfy the inequality |x-4|+|x+4|<=10, we need to break down the equation into different cases.
Case 1: When x is greater than or equal to -4
In this case, |x-4| becomes x-4, and |x+4| remains as x+4.
So the inequality becomes (x-4) + (x+4) <= 10.
Simplifying this expression, we have:
2x <= 10
x <= 5
Therefore, in this case, the values of x that satisfy the inequality are x ≤ 5.
Case 2: When x is less than -4
In this case, |x-4| becomes -x+4, and |x+4| becomes -x-4.
So the inequality becomes (-x+4) + (-x-4) <= 10.
Simplifying this expression, we have:
-2x <= 10
x >= -5
Therefore, in this case, the values of x that satisfy the inequality are x ≥ -5.
Combining both cases, we can conclude that the values of x that satisfy the original inequality |x-4|+|x+4|<=10 are -5 ≤ x ≤ 5.