solve each inequality algebraically and give the answer by using interval notation.
5-9y <_ (less than or equal to) 19-2y
I got -2y <_ (less than or equal to) y; (-2, infinity]. But the back of the book answer says [infinity, -2). What am I doing wrong?
Does it really make a difference in which order the solutions are given? You have the same solutions.
To solve the inequality algebraically, let's break it down step by step:
5 - 9y ≤ 19 - 2y
First, we can simplify the inequality by combining like terms:
-9y + 2y ≤ 19 - 5
-7y ≤ 14
Next, we need to isolate the variable y by dividing both sides of the inequality by -7. When dividing by a negative number, remember to reverse the inequality sign:
y ≥ 14 / -7
y ≥ -2
Now, we seem to have made an error in the direction of the inequality sign. Let's examine the mistake:
Instead of -2y ≤ y, it should be -2y ≥ y. This mistake occurred while dividing by -7 and reversing the inequality sign.
So the corrected inequality is:
-2y ≥ y
Now, we can solve for y:
-2y - y ≥ 0
-3y ≥ 0
Dividing by -3 and reversing the inequality sign:
y ≤ 0
Hence, the corrected solution for the inequality is:
y ≥ -2 and y ≤ 0
To express this solution in interval notation, we write it as:
[-2, 0]
Therefore, the correct answer is [-2, 0], not [∞, -2).