"Dr. Tydings has found that, over the years, 95% of babies he has delivered have weighed y pounds, where
|y-8.0|<=1.5
What range of weights corresponds to this inequality?"
|y-8.0|<=1.5
y-8 ≤ 1.5 and -y+8 ≤ 1.5
y ≤ 9.5 and -y ≤ -6.5
y ≤ 9.5 and y ≥ 6.5
which can be written as
6.5 ≤ y ≤ 9.5
(or in words, y is between 6.5 and 9.5 inclusive
Well, if we take a closer look at the inequality |y-8.0|<=1.5, we realize that it's quite the delicately balanced equation. It's like walking on a tightrope while juggling, with the weight of the babies being the juggling balls. Now, Dr. Tydings, being a skilled acrobat in the world of baby delivery, has observed that 95% of the babies he has delivered fall within this range.
So, let's crunch some numbers here. The range |y-8.0|<=1.5 basically means that the weight of the babies is within 1.5 units of 8.0, with "y" representing the weight. To stay within this range, the weights can't be too light or too heavy, just like those delicate juggling balls.
To determine the range, we can subtract and add 1.5 from 8.0. This gives us a lower bound of 6.5 (8.0 - 1.5) and an upper bound of 9.5 (8.0 + 1.5).
So, according to Dr. Tydings' observations, the range of weights that corresponds to this inequality is between 6.5 and 9.5 pounds. Just remember, if you’re planning to juggle babies, be sure to follow this weight range for an optimal juggling experience.
To find the range of weights that corresponds to the given inequality, we need to solve the inequality:
|y - 8.0| ≤ 1.5
First, let's consider the two cases when y - 8.0 is positive and negative:
1. When y - 8.0 ≥ 0:
In this case, the inequality becomes:
y - 8.0 ≤ 1.5
Solving for y, we add 8.0 to both sides of the inequality:
y ≤ 1.5 + 8.0
Simplifying, we get:
y ≤ 9.5
2. When y - 8.0 < 0:
In this case, the inequality becomes:
-(y - 8.0) ≤ 1.5
Multiplying both sides by -1 flips the inequality direction:
y - 8.0 ≥ -1.5
Solving for y, we add 8.0 to both sides of the inequality:
y ≥ -1.5 + 8.0
Simplifying, we get:
y ≥ 6.5
Therefore, the range of weights that corresponds to the inequality |y - 8.0| ≤ 1.5 is 6.5 ≤ y ≤ 9.5.
To find the range of weights that corresponds to the given inequality, we need to solve the inequality |y-8.0| ≤ 1.5. Here's how you can do that:
Step 1: Set up the inequality
|y - 8.0| ≤ 1.5
Step 2: Break down the inequality into two separate inequalities
(y - 8.0) ≤ 1.5 and -(y - 8.0) ≤ 1.5
Step 3: Solve each inequality separately
For the first inequality, add 8.0 to both sides:
y - 8.0 + 8.0 ≤ 1.5 + 8.0
y ≤ 9.5
For the second inequality, distribute the negative sign:
-y + 8.0 ≤ 1.5
Subtract 8.0 from both sides:
-y ≤ 1.5 - 8.0
-y ≤ -6.5
Multiply both sides by -1 (since we are multiplying by a negative number, we need to reverse the inequality symbol):
y ≥ 6.5
Step 4: Combine the results
Since we have two inequalities, y ≤ 9.5 and y ≥ 6.5, the range of weights that correspond to the given inequality is 6.5 ≤ y ≤ 9.5.
Therefore, the range of weights is from 6.5 to 9.5 pounds (inclusive).