If |3x 19| 5 x , which of the following represents the complete

Question

If |3x – 19| > 5 – x , which of the following represents the complete range of values of x ?

A. x > 6
B. x < 7
C. 6 < x < 7
D. All real values
E. No real value

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Official Answer

D

The_ashura · Accepted Answer

So we have |3x-19|>5-x
Or
-5+x>3x-19>5-x
+5 from the equation
x>3x-14>10-x

Split for ease.
1st-
x>3x-14 => 2x<14 => x<7

2nd-
3x-14>10-x => 4x>24 => x<6

Now, How does this look like?
X<6 and x>7?

Yep.
If you combine these two directions on the number line, they completely overlap and cover every number from negative infinity to positive infinity.

Any number less than 7 (like 0 or -10) satisfies Case 2.
Any number greater than 6 (like 10 or 100) satisfies Case 1.
Numbers between 6 and 7 (like 6.5) satisfy both cases.

Therefore, every single real number will satisfy at least one of the two conditions.
Hence, D.All real numbers.

SotJake · Answer

Since the question asks for the  complete  range we can check if the numbers that are excluded by some of the answer choices actually are in this range.

We try  x=6  and the inequality becomes: 
 |18-19|>5-6 \implies 1>-1 , which is true.
Thus, we can eliminate  (A) ,  (C)  and  (E) .

We try  x=7  and the inequality becomes: 
 |21-19|>5-7 \implies 2>-2 , which is true.
Thus, we can also eliminate  (B) .

Correct answer:  (D)

Ranjit.C · Answer

So, this can be done by traditional algebra. However, solving such inequalities in most cases can be done by option elimination as well.

|3x – 19| > 5 – x

Check if the inequality holds true for x = 0
LHS => |3(0) - 19| = 19 
RHS => 5 - 0 = 5

x = 0 satisfies the inequality, hence should be a part of the solution.

Hence, we can eliminate options - A, C, E

Now choose a value of x which is in option 4 but not in option 1.

Let's say x = 10

LHS => |3(10) - 19| = 11
RHS => 5 - 10 = -5
x = 10 satisfies the inequality, hence should be a part of the solution. Hence, option B is eliminated.

Answer = option D

ExpertsGlobal5 · Answer

D is the correct answer choice.

Video explanation:

https://www.youtube.com/watch?v=dUtI2VA29Wg

If |3x 19| > 5 x , which of the following represents the complete

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