A landscaping company sells 40-pound bags of top soil. The actual weight x of a bag, however, may differ from the advertised weight by as much as 0.75 pound. Write an inequality involving absolute value that expresses the relationship between the actual weight x of a bag and 40 pounds. Solve the inequality, and express the answer in interval form.
|x - 40| < 0.75
or
39.25 < x < 40.75
Thank you.. I wasn't for sure if I had to add x+40 or -, but I see now.
Sure, let me clown around with the problem for a moment!
To express the relationship between the actual weight x of a bag and 40 pounds with absolute value, we can use the following inequality:
| x - 40 | ≤ 0.75
Now let's solve it without any clowning around. When we remove the absolute value, we'll get two separate inequalities:
x - 40 ≤ 0.75 AND -(x - 40) ≤ 0.75
Simplifying each inequality, we have:
x ≤ 40.75 AND -x + 40 ≤ 0.75
Now let's solve for x in each inequality:
For x ≤ 40.75, x can go as low as negative infinity up to 40.75, so the interval is (-∞, 40.75].
For -x + 40 ≤ 0.75, we subtract 40 from each side and multiply by -1, giving us:
x - 40 ≥ -0.75
x ≥ 40 - 0.75
x ≥ 39.25
So, for x ≥ 39.25, x can go as high as positive infinity or equal to or above 39.25, so the interval is [39.25, ∞).
Since we're looking for the weight of the bag, we need to find the common solution between the two intervals, which is [39.25, 40.75].
Voila! There you go, an interval form answer without any clowning around!
The inequality involving absolute value that expresses the relationship between the actual weight x of a bag and 40 pounds is:
| x - 40 | ≤ 0.75
To solve this inequality, we need to consider two cases:
Case 1: x - 40 ≥ 0 (when x is greater than or equal to 40)
In this case, the inequality simplifies to:
x - 40 ≤ 0.75
x ≤ 40 + 0.75
x ≤ 40.75
Case 2: x - 40 < 0 (when x is less than 40)
In this case, the inequality simplifies to:
-(x - 40) ≤ 0.75
- x + 40 ≤ 0.75
- x ≤ 0.75 - 40
- x ≤ -39.25
x ≥ 39.25
Combining the solutions from both cases, we get the interval solution:
x ∈ [39.25, 40.75]
To write the inequality involving absolute value, we need to consider the difference between the actual weight x and the advertised weight 40 pounds:
|x - 40| ≤ 0.75
This inequality states that the absolute value of the difference between x and 40 is less than or equal to 0.75.
To solve the inequality, we can consider two cases:
Case 1: (x - 40) ≥ 0
In this case, the absolute value sign is not needed, so we can rewrite the inequality as:
x - 40 ≤ 0.75
Solving for x gives:
x ≤ 40.75
Case 2: (x - 40) < 0
In this case, we need to multiply both sides of the inequality by -1:
-(x - 40) ≤ 0.75
Distributing the negative sign:
- x + 40 ≤ 0.75
Subtracting 40 from both sides:
- x ≤ -39.25
Dividing both sides by -1 (and reversing the inequality symbol):
x ≥ 39.25
Therefore, the solution to the inequality |x - 40| ≤ 0.75 is the interval [39.25, 40.75]. This interval represents all the possible values for the actual weight x of a bag that are within 0.75 pounds of the advertised weight 40 pounds.