you are building a rock wall for a flower bed along your 38-foot long driveway. You Want the wall to be one-half of the length of the driveway with an absolute deviation of at most 3 feet. write and solve an absolute value inequality that represents the possible lengths of the wall
Let's call the length of the rock wall "x". According to the given information, we want the wall to be one-half of the length of the driveway, which is 38 feet.
So, we have the equation:
x = (1/2) * 38
Simplifying, we get:
x = 19
Now, we need to incorporate the absolute deviation of at most 3 feet into our equation. To represent this using an absolute value inequality, we can write:
|x - 19| ≤ 3
This inequality states that the distance between x and 19 should be less than or equal to 3. By solving this inequality, we can find the possible lengths of the wall.
To solve this problem, we need to remember that the absolute value of a number is always positive.
Let's define the length of the wall as "x". According to the problem, the wall should be one-half the length of the driveway, which is 38 feet. So we can write this as an equation:
x = (1/2) * 38
Simplifying this equation, we have:
x = 19
To represent the absolute deviation, we need to consider that the wall's length can deviate by at most 3 feet from this length, both below and above.
To calculate the lower limit, we subtract 3 feet from the desired length:
x - 3 ≥ 19
To calculate the upper limit, we add 3 feet to the desired length:
x + 3 ≤ 19
Combining these inequalities, we have:
x - 3 ≥ 19 or x + 3 ≤ 19
Simplifying each inequality, we get:
x ≥ 22 or x ≤ 16
Therefore, the absolute value inequality that represents the possible lengths of the wall is:
| x - 19 | ≤ 3
This means that the length of the wall can be any value between 16 and 22 feet, inclusive.