Complete step-by-step solutionWhat is the value of \(|x + 5| + |x - 3|\) ?The critical points (aka key points or transition points) are -5 and 3 (the values of x for which the expressions in the absolute values become 0).
Consider three ranges:
- If \(x < - 5\), then \(x + 5 < 0\) and \(x - 3 < 9\), so \(|x + 5| = -(x + 5)\) and \(|x - 3| = -(x - 3)\). Thus in this range \(|x + 5| + |x - 3|\) becomes \(-(x + 5) - (x - 3) = -2 - 2x\).
- If \(- 5 \leq x \leq 3\), then \(x + 5 \geq 0\) and \(x - 3 \leq 9\), so \(|x + 5| = x + 5\) and \(|x - 3| = -(x - 3)\). Thus in this range \(|x + 5| + |x - 3|\) becomes \(x + 5 - (x - 3) = 8\).
- If \(x > 3\), then \(x + 5 > 0\) and \(x - 3 > 9\), so \(|x + 5| = x + 5\) and \(|x - 3| = x - 3\). Thus in this range \(|x + 5| + |x - 3|\) becomes \(x + 5 + (x - 3) = 2x + 2\).
The above mean that if x is in the first range (\(x < - 5\)) or in the third range (\(x > 3\)), then the value of \(|x + 5| + |x - 3|\) depends on the value of x. For example:
If \(x = -10\), then \(|x + 5| + |x - 3|= -2 - 2x=18\);
If \(x = -7\), then \(|x + 5| + |x - 3|= -2 - 2x=12\);
If \(x = 4\), then \(|x + 5| + |x - 3|= 2x + 2=10\);
If \(x = 6\), then \(|x + 5| + |x - 3|= 2x + 2=14\).
But if x is in the second range (\(- 5 \leq x \leq 3\)), then the value of \(|x + 5| + |x - 3|\) is independent of the value of x, and is ALWAYS equals to 8. For example:
If \(x = -5\), then \(|x + 5| + |x - 3|= 8\);
If \(x = 0\), then \(|x + 5| + |x - 3|= 8\);
If \(x = 3\), then \(|x + 5| + |x - 3|= 8\).
(1) \(x^2< 25\):
Take the square root: \(|x| < 5\);
Get rid of the absolute value sign: \(-5 < 0 < 5\);
x can be in the second or third range. So, \(|x + 5| + |x - 3|\) is either 8 or \(2x + 2\). Not sufficient.
(2) \(x^2 > 9\):
Take the square root: \(|x| > 3\);
Get rid of the absolute value sign: \(x < -3\) or \(x > 3\);
x can be in any of the three ranges from above. So, \(|x + 5| + |x - 3|\) is \(-2 - 2x\), 8 or \(2x + 2\). Not sufficient.
(1)+(2) We get \(-5 < x < -3\) (second range) or \(3 < x < 5\) (third range). If \(-5 < x < -3\) (second range), then \(|x + 5| + |x - 3|=8\) but if \(3 < x < 5\) (third range), then \(|x + 5| + |x - 3|=2x + 2\) (so the value will depend on the exact value of x). Not sufficient.
Answer: E.
Can you please explain why you are checking if (x-3) is lesser than or greater than 9 and not 0 in each of the three ranges?