(05.05 lc) determine whether the origin is included in the shaded region and whether the shaded region is above or below the line for the graph of the following inequality: y < two thirds x + 2 the origin is not included in the shaded region and the shaded area is above the line. the origin is not included in the shaded region and the shaded area is below the line. the origin is included in the shaded region and the shaded area is above the line. the origin is included in the shaded region and the shaded area is below the line

The origin is (0,0)

is it included in the shaded regionanother words, is it one solution to ur inequality...so we sub to find out...

y < 2/3x + 2subbing in (0,0)...x = 0 and y = 0
0 < 2/3(0) + 2
0 < 2 (this is correct)so (0,0), the origin, is included in the shaded region...and since ur inequality sign is < (less then), it is shaded below the line

Intersecting secant-tangent theorem states that if a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external secant segment.so to break it down: 1. pq acts as the tangent and ps as the secant with an intersection at p.sr = 21rp = 3x + 3pq = 4x + 4following to equation: tangent squared = the two segments multiplied by each otherso (4x + 4) ^ 2 = (3x + 3) • (21) simplifies to 16x^2 - 31x - 47 = 0 or (16x - 47)(x + 1) x = -1 or x = 47/162. pq acts as the tangent and ps as the secant with an intersection at p (again! )sr = 16rp = xpq = 15(15) ^ 2 = 16 • x225 = 16xx = 225/163. the interesting chords are proportional to each other so a ratio is possible to set up: ap dp 10 3 + x = = —— = pc pb 8 xcross multiple to get 10x = (3 + x)(8)x = 12