Question

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. (x+3)(x−5)\u003e0

Accepted Answer

Start by identifying the critical points of the inequality $$(x+3)(x-5) \u003e 0. $$These are the values of $$x$$ that make each factor equal to zero, so solve $$x+3=0$$ and $$x-5=0 to $$find the critical points. The critical points divide the real number line into intervals. In this case, the intervals are $$(-\infty, -3)$$, $$(-3, 5)$$, and $$(5, \infty). We $$will test each interval to determine where the product $$(x+3)(x-5) is $$positive. Choose a test point from each interval and substitute it into the expression $$(x+3)(x-5). $$For example, pick $$x = -4$$ for $$(-\infty, -3)$$, $$x = 0$$ for $$(-3, 5)$$, and $$x = 6$$ for $$(5, \infty). $$Determine the sign of the product at each test point. Based on the signs from the test points, identify which intervals satisfy the inequality $$(x+3)(x-5) \u003e 0. $$Remember, the inequality is strict, so the critical points themselves are not included in the solution set. Express the solution set as a union of intervals where the product is positive, and then graph this solution on a real number line, marking open circles at the critical points $$x = -3$$ and $$x = 5 to $$indicate they are not included.

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. (x+3)(x−5)>0

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Polynomial Inequalities

Critical Points and Sign Analysis

Interval Notation and Graphing on the Number Line