Questions: Solve the polynomial inequality and graph the solution set on a number line. Express the solution set in interval notation. x^2-12x+36>0

Solution Steps

To solve the polynomial inequality \(x^2 - 12x + 36 > 0\), we first find the roots of the corresponding equation \(x^2 - 12x + 36 = 0\). This will help us determine the intervals to test for the inequality. After finding the roots, we test intervals around these roots to see where the inequality holds true. Finally, we express the solution set in interval notation.

The given polynomial inequality is:

\[ x^2 - 12x + 36 > 0 \]

First, we need to factor the quadratic expression \(x^2 - 12x + 36\). Notice that this is a perfect square trinomial:

\[ x^2 - 12x + 36 = (x - 6)^2 \]

We need to solve the inequality:

\[ (x - 6)^2 > 0 \]

The expression \((x - 6)^2\) is a square of a binomial, which is always non-negative. It equals zero when \(x = 6\). Therefore, \((x - 6)^2 > 0\) for all \(x\) except \(x = 6\).

Since \((x - 6)^2 > 0\) for all \(x \neq 6\), the solution set is all real numbers except \(x = 6\).

The solution set in interval notation is:

\[ (-\infty, 6) \cup (6, \infty) \]

To graph the solution set on a number line:

• Draw a number line.
• Place an open circle at \(x = 6\) to indicate that this point is not included in the solution set.
• Shade the regions to the left of \(x = 6\) and to the right of \(x = 6\) to indicate that all other real numbers are included in the solution set.

Final Answer