Answer: The diagram is shown below
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Explanation:
If you graphed y = (x-5)(x+5) on the xy grid, then you'd see a parabola forming. This parabola has x intercepts at -5 and 5. The parabola opens upward.
If x < -5, then (x-5)(x+5) is positive. Let's say x = -6
(x-5)(x+5)
(-6-5)(-6+5)
(-11)(-1)
11
So if x = -6, then y = (x-5)(x+5) becomes 11 which is positive. If you picked any other value less than -5, then (x-5)(x+5) is positive.
Similarly, if x > 5, then (x-5)(x+5) is also positive for similar reasoning. Try x = 6 to see what happens. The actual value itself doesn't matter. All we care about is if the result is positive or negative.
It's only when -5 < x < 5 that (x-5)(x+5) is negative. Consider x = 0 which is between -5 and 5
Plugging x = 0 leads us to
(x-5)(x+5)
(0-5)(0+5)
(-5)(5)
-25
Showing that (x-5)(x+5) is negative on the interval -5 < x < 5
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Going back to the original inequality, we want to know when (x-5)(x+5) is positive or 0
We found that (x-5)(x+5) is positive when x < -5 or when x > 5
(x-5)(x+5) is equal to zero when x = -5 or x = 5
So overall, the solution we want is [tex]x \le -5 \ \text{ or } \ x \ge 5[/tex]
To graph this on a number line, we plot closed filled in circles at -5 and 5. Then we shade everything that is not between these two circles (so we shade everything except the interval -5 < x < 5).
The graph is below.
