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x^2 is less than or equal to 25

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2 Answers

X^2 less than or equal to 25, X is less than or equal to 5.  (5)^2 = 25, and (-5)^2 = 25, so the solution is any value of x between, and including, 5, -5.  If you were to plot this on a number line, you would have a "solid" or "bubbled in" circle on -5 and +5 and a line between them.  The interval notation would be [-5,5]... since you are including both 5 and -5 in the solution, you have to use brackets, not parenthesis.

The reason you INCLUDE -5 and 5 is because the equation says "less than OR equal to"  the phrase "equal to" indicates that you include both numbers in the solution.

Let's remember that equation
x2 = a (were a ≥ 0 and "x" is real number) has 2 roots:
√(±x)2 = √a , so ±x = √a , x = ±√a , or lxl = √a
To solve the inequality x2 ≤ 25 (52 = 25 and (-5)2 = 25) we have to examine the inequality on two intervals/segments:
1. if 0 ≤ x ≤ 5 (x is positive, or equal 0) , then x ≤ 5
2. if -5 ≤ x < 0 (x is negative) , -x ≤ 5 (when we multiply/divide inequality by negative number, we have to switch sign of inequality to opposite) -x/ (-1) ≥ 5/(-1) ------> x ≥ -5
So the answer is -5 ≤ x ≤ 5 , or any number from the segment [-5,5]