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−4)(x+2)\u003e0

Accepted Answer

Start by identifying the critical points of the inequality $$(x-4)(x+2) \u003e 0. $$These are the values of $$x$$ that make each factor equal to zero, so set each factor equal to zero: $$x - 4 = 0$$ and $$x + 2 = 0. $$Solve each equation to find the critical points: $$x = 4$$ and $$x = -2. $$These points divide the real number line into three intervals: $$(-\infty, -2)$$, $$(-2, 4)$$, and $$(4, \infty). $$Test a value from each interval in the original inequality $$(x-4)(x+2) \u003e 0 to $$determine if the product is positive or negative in that interval. For example, pick $$x = -3$$ for $$(-\infty, -2)$$, $$x = 0$$ for $$(-2, 4)$$, and $$x = 5$$ for $$(4, \infty). $$Determine the sign of the product in each interval by substituting the test values into $$(x-4)(x+2). If $$the product is greater than zero, that interval is part of the solution set; if not, it is excluded. Express the solution set as the union of intervals where the inequality holds true, and write it in interval notation. Also, graph these intervals on the real number line, using open circles at the critical points since the inequality is strict (greater than zero, not greater than or equal to).

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−4)(x+2)>0

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Concetti chiave

Polynomial Inequalities

Critical Points and Sign Analysis

Interval Notation and Graphing on the Number Line