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Chapter 2: Problem 19

Solve each polynomial inequality and graph the solution set on a real number line. Express each solution set in interval notation. $$x^{2}-4 x \geq 0$$

Short Answer

Expert verified
The solution set for the given inequality in interval notation is \([0,0]\cup[4,\infty)\). The inequality holds true for \(x=0\) and \(x \geq 4\).

Step by step solution

01

Find the critical values

To find the critical values, rewrite the inequality into a quadratic equation and solve for \(x\). Thus, we have \(x^2 - 4x = 0\). The quadratic can be factored to \(x(x-4) = 0\). By setting each factor equal to zero, we find the critical values \(x=0\) and \(x=4\).
02

Construct a Sign Chart

A sign chart helps determine the regions where the inequality holds true. We have three regions to test because we have two critical values: \(x<0\), \(04\). Test these intervals with the inequality \(x^2 - 4x \geq 0\). For \(x<0\), substitute \(-1\) into the inequality (hence \(1<0\), which is false). For \(04\), substitute \(5\) into the inequality (giving \(5\geq0\), which is true).
03

Express in Interval Notation and Graph

Interval notation for \(x\geq4\) would be \([4,\infty)\). Together with \(x=0\), we have \([0,0]\cup[4,\infty)\). The graph is a closed circle at \(x=0\) and \(x=4\), with a line extended to \(\infty\) from \(x=4\).

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