Solving Absolute Value Inequalities with a Non-Negative Constant
This inequality involves an absolute value. The statement |x| ≥ k, where k is a non-negative number, is equivalent to saying that x ≤ -k or x ≥ k, since the absolute value indicates the distance of a number from zero, not the direction.
Given the inequality:
50 ≤ |1 - g + 950|
we can split this into two separate inequalities because the expression inside the absolute value must either be greater than or equal to 50 or less than or equal to -50.
Therefore, we have:
1. 1 - g + 950 ≥ 50
2. 1 - g + 950 ≤ -50
Let's solve each inequality.
For the first inequality:
1 - g + 950 ≥ 50
Subtract 950 from both sides:
1 - g ≥ 50 - 950
1 - g ≥ -900
Now, add g to both sides:
1 ≥ g - 900
Finally, add 900 to both sides:
g ≤ 901
For the second inequality:
1 - g + 950 ≤ -50
Subtract 950 from both sides:
1 - g ≤ -50 - 950
1 - g ≤ -1000
Add g to both sides:
1 + g ≤ -1000
Add 1 to both sides:
g ≤ -1001
Combining both sets of g values, we have:
g ≤ 901 and g ≤ -1001
However, since g cannot be simultaneously less than or equal to both 901 and -1001, we consider the entire range of g that satisfies either condition. Thus, the solution to the inequality is:
g ≤ 901
Since -1001 is much less than 901, the latter encompasses all possible values of g that could satisfy the inequality.
