Question - Solving Absolute Value Inequalities with a Non-Negative Constant
Solution:
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 ≥ 502. 1 - g + 950 ≤ -50Let's solve each inequality.For the first inequality:1 - g + 950 ≥ 50Subtract 950 from both sides:1 - g ≥ 50 - 9501 - g ≥ -900Now, add g to both sides:1 ≥ g - 900Finally, add 900 to both sides:g ≤ 901For the second inequality:1 - g + 950 ≤ -50Subtract 950 from both sides:1 - g ≤ -50 - 9501 - g ≤ -1000Add g to both sides:1 + g ≤ -1000Add 1 to both sides:g ≤ -1001Combining both sets of g values, we have:g ≤ 901 and g ≤ -1001However, 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 ≤ 901Since -1001 is much less than 901, the latter encompasses all possible values of g that could satisfy the inequality.
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