My second question is in section 5.4, problem 39.
x+4/x-2 < or = 1; Since the inequality in not , should the answer to this problem in the back of the book be: {x|x< or = 2}, or in interval notation: (-infinity, 2)? Please explain this problem. Thanks.
To solve the inequality x + 4/(x - 2) ≤ 1, first, we need to find the values of x that satisfy the inequality. Here's how to approach the problem:
Step 1: Remove the fraction by finding a common denominator. The denominator of the fraction is (x - 2), so multiply both sides of the inequality by (x - 2) to get rid of the fraction:
(x + 4) ≤ (x - 2)
Step 2: Simplify the inequality. Distribute the (x - 2) on the right side:
x + 4 ≤ x - 2
Step 3: Simplify further. Since we want to isolate x, move all x terms to one side:
x - x + 4 ≤ -2
4 ≤ -2
Step 4: Analyze the result. The inequality 4 ≤ -2 is not true; therefore, there is no solution to the inequality x + 4/(x - 2) ≤ 1. This means that there are no values of x that make the inequality true.
Therefore, the answer to this problem should be an empty set: {}. In interval notation, it would be the empty set, which is represented as (∅) or the set with no numbers.