Gen Math MODULE 4 - Rational Inequality PDF

1 2 3 4 5 6 7 8  2𝑥 ≥1 𝑥+1 2𝑥 −1≥0 𝑥+1 2𝑥 − (𝑥 + 1) ≥0 𝑥+1 2𝑥 − 𝑥 − 1 ≥0 𝑥+1 𝑥−1 ≥0 𝑥+1 9  2𝑥 ≥1 𝑥+1 𝑥−1 ≥0 𝑥+1 1−1 0 = 1+1 2 −1 −...

1 2 3 4 5 6 7 8  2𝑥 ≥1 𝑥+1 2𝑥 −1≥0 𝑥+1 2𝑥 − (𝑥 + 1) ≥0 𝑥+1 2𝑥 − 𝑥 − 1 ≥0 𝑥+1 𝑥−1 ≥0 𝑥+1 9  2𝑥 ≥1 𝑥+1 𝑥−1 ≥0 𝑥+1 1−1 0 = 1+1 2 −1 − 1 −2 = −1 + 1 0 Mark these on a number line. Use a shaded circle for x = 1 (part of the solution) and an unshaded circle for x = -1 (not a solution). -1 1 10  2𝑥 ≥1 𝑥+1 𝒙 < −𝟏 −𝟏 < 𝒙 < 𝟏 𝒙>𝟏 -1 1 Choose convenient test points in the intervals determined by -1 and 1 to 𝑥−1 determine the sign of in these intervals. 𝑥+1 Since we are looking for the intervals where the fraction is positive or zero, we determine the solution intervals to be 𝒙 < −𝟏 and 𝒙 ≥ 𝟏. 11  2𝑥 ≥1 𝑥+1 Plot these intervals on the number line. -1 1 Solution Set: {𝑥 ∈ ℝ|𝑥 < −1 𝑜𝑟 𝑥 ≥ 1} Interval Notation: −∞, −1 ∪ [1, ∞) 12 13 14 15 16 Examples: 𝒙+𝟒 Solve ≤ 𝟑 and graph the solution set of the inequality. 𝒙−𝟐 17 18 19 20 21