write the verbal sentence as an inequality and solve the inequality. (this has to do with absolute value inequalities)
the distance between x and 8 is less than 14
i think the answer is (x-8)<14
the parentheses stand for the absolute value sign
| x - 8 | < 14
this means
x - 8 < 14
x < 22
-x + 8 < 14
-x < 6
x > -6
therefore,
-6 < x < 22
To write the verbal sentence as an inequality, we can start by translating "the distance between x and 8 is less than 14" into mathematical symbols.
The distance between two numbers can be expressed using the absolute value. So, the distance between x and 8 can be written as |x - 8|.
Based on the sentence, we are looking for cases where this distance is less than 14. Thus, the inequality can be written as:
|x - 8| < 14
To solve this inequality, we can break it down into two separate cases:
Case 1: (x - 8) < 14
In this case, we can simply solve for x by isolating it:
x - 8 < 14
x < 14 + 8
x < 22
Case 2: -(x - 8) < 14
To simplify this, we need to distribute the negative sign:
-x + 8 < 14
Next, we isolate x by moving the 8 to the other side of the inequality:
-x < 14 - 8
-x < 6
Since we are multiplying by -1, it is necessary to flip the inequality sign:
x > -6
Combining the solutions from the two cases, we have:
-6 < x < 22
So, any value of x that falls within the range of -6 to 22 (excluding the endpoints) will satisfy the inequality |x - 8| < 14.