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Section 2.1 : Solutions and Solution Sets

5. Is \(z = 4\) a solution to \(6z - {z^2} \ge {z^2} + 3\)?

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There really isn’t all that much to do for these kinds of problems. All we need to do is plug the given number into both sides of the inequality and check to see if the inequality is true. In this case that will mean checking to see if the left side is larger than or equal to the right side.

Here is that work for this particular problem.

\[\begin{align*}6\left( 4 \right) - {\left( 4 \right)^2} & \mathop \ge \limits^? {\left( 4 \right)^2} + 3\\ 8\require{cancel} & \bcancel{ \ge }19\,\,\,\,{\mbox{NOT OK}}\end{align*}\]

So, we can see that the left side is neither larger than nor equal to the right side and so \(z = 4\) is not a solution to this inequality.