Questions: Graph the solution to the following inequality on the number line. x^2-8x>-12

Solution Steps

We are asked to graph the solution to the inequality \(x^2 - 8x > -12\) on the number line. First, we rewrite the inequality as follows: \[x^2 - 8x + 12 > 0\]

We factor the quadratic expression \(x^2 - 8x + 12\) as \((x-2)(x-6)\). So, the inequality becomes \[(x-2)(x-6) > 0\]

The critical points are the values of \(x\) that make the expression equal to 0. In this case, the critical points are \(x=2\) and \(x=6\).

We test the intervals determined by the critical points: \(x<2\), \(2<x<6\), and \(x>6\).

• If \(x<2\), for instance \(x=0\), then \((0-2)(0-6) = (-2)(-6) = 12 > 0\). So, the inequality holds for \(x<2\).
• If \(2<x<6\), for instance \(x=4\), then \((4-2)(4-6) = (2)(-2) = -4 < 0\). So, the inequality does not hold for \(2<x<6\).
• If \(x>6\), for instance \(x=8\), then \((8-2)(8-6) = (6)(2) = 12 > 0\). So, the inequality holds for \(x>6\).

The solution set is \(x<2\) or \(x>6\). In interval notation, this is \((-\infty, 2) \cup (6, \infty)\).

The graph should show open circles at \(x=2\) and \(x=6\), and the intervals to the left of 2 and to the right of 6 should be shaded.

Final Answer