Suppose that C and D are points on the number line. If =CD4 and D lies at −12, where could C be located?
If "=CD4" means "CD=4" then C will be at either
-12 + 4 = -8
or
-12 - 4 = -16
That is, the distance between -12 and C is 4
|-12-C| = 4
To determine the possible locations of point C, we need to find the range of values that satisfy CD = 4 and D = -12.
Since CD represents the distance between points C and D on the number line, we can express it as the absolute value of their difference: |C - D|.
Given that CD = 4, we can write the equation as:
|C - (-12)| = 4
Simplifying this equation, we get:
|C + 12| = 4
To solve for C, we need to consider two cases, depending on whether C + 12 is positive or negative.
Case 1: C + 12 is positive
If C + 12 is positive, then we can remove the absolute value signs:
C + 12 = 4
Solving for C:
C = 4 - 12
C = -8
So, in this case, if C + 12 is positive, then C = -8.
Case 2: C + 12 is negative
If C + 12 is negative, then the absolute value of a negative number is its positive counterpart. Thus, we can remove the absolute value signs by negating the expression inside:
-(C + 12) = 4
Simplifying:
-C - 12 = 4
Solving for C:
-C = 4 + 12
-C = 16
Multiplying both sides by -1 to isolate C:
C = -16
So, in this case, if C + 12 is negative, then C = -16.
Therefore, the possible locations of point C, given CD = 4 and D = -12, are C = -8 or C = -16 on the number line.