Types Of Sets

Define We want to find all for which has exactly one real solution. Observe that are both piecewise linear but each is non-decreasing in certain intervals and non-decreasing overall when appropriately pieced together. A more detailed analysis (or plotting) shows that is strictly increasing as a function of . Since is strictly increasing in , the equation intersects the -axis exactly once, regardless of the value of . Consequently, for any real , there is exactly one solution to . Hence, the set of all such is

Step 1: Form the augmented matrix of the system
The given system of equations can be written as the augmented matrix: For the system to have no solution, the determinant of the coefficient matrix must be zero.
Step 2: Calculate the determinant We calculate the determinant : Simplifying the 2x2 determinants: Expanding the terms:
Step 3: Solve for
For the system to have no solution, the determinant must be zero: Divide the entire equation by 12:
Step 4: Find the roots of the cubic equation Solving the cubic equation by using trial and error or factoring, we find the roots:
Step 5: Calculate
Now, the values of are . The sum of their absolute values is: Thus, Thus, the required value is .