1/2|6y-9| +5 is < or = 2
i solved it and got two fractions but the answer is 0 can anyone explain?
btw this is absolute value inequality.
1/2|6y-9|+5 <= 2
1/2|6y-9|<=-3
multiply by 2 making the signs switch
|6y-9|>=-6
Since it's absolute value, do this:
6>=6y-9>=-6
15>=6y>=3
5/2 >= y >= 1/2
Jen was right up to where she multiplied by +2 and switched the inequality sign.
the sign only switches when multiplying or dividing by a negative number so it should have been
|6y-9| ≤ -6
at this point we have a "violation" of the definition of absolute value.
by definition the absolute value of anything has to be ≥ 0
So there is no solution.
Lyne , you said the answer was zero, it is not.
try subbing it in, it won't work, no number works.
ohh okay thanks so much that makes sense!
thanks again =)haha
To solve the absolute value inequality 1/2|6y-9| + 5 ≤ 2, we will break it down into two separate cases: when the expression inside the absolute value is positive and when it is negative.
Case 1: 6y - 9 ≥ 0
We solve the equation 6y - 9 = 0 by adding 9 to both sides, which gives 6y = 9. Dividing both sides by 6, we obtain y = 3/2. This is the critical point where the expression inside the absolute value changes from positive to negative.
Case 2: 6y - 9 < 0
We solve the inequality 6y - 9 < 0 by adding 9 to both sides, resulting in 6y < 9. By dividing both sides by 6, we get y < 3/2.
Now, let's evaluate the original inequality for both cases:
Case 1: When y ≥ 3/2
Substituting y = 3/2 into the expression, we have 1/2|6(3/2) - 9| + 5 ≤ 2.
Simplifying further, we get 1/2|9 - 9| + 5 ≤ 2.
Since the absolute value of 0 is 0, we have 1/2(0) + 5 ≤ 2.
Therefore, 5 ≤ 2, which is not true, so this case does not satisfy the inequality.
Case 2: When y < 3/2
Substituting y = 1 into the expression, we have 1/2|6(1) - 9| + 5 ≤ 2.
Simplifying further, we get 1/2|6 - 9| + 5 ≤ 2.
Since the absolute value of -3 is 3, we have 1/2(3) + 5 ≤ 2.
Therefore, 3/2 + 5 ≤ 2.
Combining like terms, we have 9/2 ≤ 2.
Now, in order for this inequality to be true, the left side must be less than or equal to the right side. However, 9/2 is greater than 2, so this case also does not satisfy the inequality.
Since neither case satisfies the inequality, there are no possible solutions for the given absolute value inequality. Hence, the answer is 0.