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.
Well, this is a classic case of weighty matters. To express the relationship between the actual weight x of a bag and 40 pounds, while taking into account the possible discrepancy of 0.75 pounds, we can use absolute value.
The inequality to represent this situation is:
| x - 40 | ≤ 0.75
Now, let's solve this inequality. When solving an absolute value inequality, we typically split it into two separate inequalities, one positive and one negative. So we'll consider both cases.
For the positive inequality:
x - 40 ≤ 0.75
If we add 40 to both sides, we get:
x ≤ 40.75
For the negative inequality:
-(x - 40) ≤ 0.75
Here, we distribute the negative sign:
- x + 40 ≤ 0.75
If we subtract 40 from both sides, we get:
- x ≤ -39.25
To isolate x, we need to multiply both sides of the inequality by -1, but since we're multiplying by a negative number, we must reverse the inequality sign:
x ≥ 39.25
So, the solution in interval form is:
[39.25, 40.75]
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 the inequality, we can consider both cases:
Case 1: x - 40 ≤ 0.75
Adding 40 to both sides:
x ≤ 40 + 0.75
x ≤ 40.75
Case 2: -(x - 40) ≤ 0.75 (Note: we take the negative sign because we are considering the possibility that the actual weight x can be less than 40)
Multiplying through by -1:
x - 40 ≥ -0.75
Adding 40 to both sides:
x ≥ 40 - 0.75
x ≥ 39.25
So, the solution is:
39.25 ≤ x ≤ 40.75
In interval form, the answer is [39.25, 40.75].
To write an inequality that expresses the relationship between the actual weight x of a bag and 40 pounds, considering the difference allowed of 0.75 pounds, we can use absolute value.
The absolute value of the difference between the actual weight and the advertised weight should be less than or equal to 0.75 pounds. Mathematically, this can be represented as:
| x - 40 | ≤ 0.75
To solve this inequality, we can consider the two possible cases:
Case 1: x - 40 ≤ 0.75
In this case, we add 40 to both sides of the inequality:
x - 40 + 40 ≤ 0.75 + 40
x ≤ 40.75
Case 2: x - 40 ≥ -0.75
In this case, we add 40 to both sides of the inequality:
x - 40 + 40 ≥ -0.75 + 40
x ≥ 39.25
So, combining both cases, the solution to the inequality in interval form is:
39.25 ≤ x ≤ 40.75