Sets and absolute value

The midpoint coordinate is the average of the endpoint coordinates. Distance is the absolute value of the coordinate difference; a negative coordinate does not mean a negative distance.

The intersection A ∩ B contains elements in both sets. Count shared elements once in the union: |A ∪ B| = |A| + |B| − |A ∩ B|. Thus intersection size = |A| + |B| − |A ∪ B|, and |A \ B| = |A| − |A ∩ B|.

Convert the first inequality into a double inequality using distance. Intersect the two conditions. Check the inclusion of each endpoint separately.

Interpret the absolute values as distances to three points. Consider the position of x relative to those points and determine where the sum of distances is smallest.

Split |kx + b| = r into two equations: kx + b = r and kx + b = −r. Solve both equations and give both solutions.

The intersection A ∩ B contains elements in both sets. Count shared elements once in the union: |A ∪ B| = |A| + |B| − |A ∩ B|. Thus intersection size = |A| + |B| − |A ∪ B|, and |A \ B| = |A| − |A ∩ B|.

Use the identity √(u²) = |u|. If −a < x < a, then x − a < 0 and x + a > 0. Therefore |x − a| = a − x and |x + a| = x + a. Add the expressions to obtain the final numerical answer.

First calculate the expressions inside the two absolute-value signs separately. Then apply the definition of absolute value: a positive number remains unchanged, while a negative number is replaced by its positive opposite. Finally add the two non-negative values.

The equation |x − a| = d means that point x is d units away from point a on the number line. Therefore it normally has two solutions: x = a − d and x = a + d.

The intersection A ∩ B contains elements in both sets. Count shared elements once in the union: |A ∪ B| = |A| + |B| − |A ∩ B|. Thus intersection size = |A| + |B| − |A ∪ B|, and |A \ B| = |A| − |A ∩ B|.