Which of the following inequalities is true?
A. -3x > 5x
B. |3x| < |2x|
C. 4x < -2x
D. -8x > 6x
To determine which of the given inequalities is true, we can solve each one step by step.
A. -3x > 5x
To find the solution for this inequality, we need to isolate the variable x. Let's add 3x to both sides: -3x + 3x > 5x + 3x. Simplifying gives 0 > 8x. Dividing both sides by 8 (and remembering to reverse the inequality because we're dividing by a negative value) gives 0 < x, or x > 0.
B. |3x| < |2x|
The absolute value inequalities involve considering both the positive and negative values of the expressions inside the absolute value signs. Let's solve this inequality by considering both cases separately.
Case 1: 3x < 2x
Subtracting 2x from both sides gives x < 0.
Case 2: -3x < 2x
Adding 3x to both sides gives 0 < 5x. Dividing both sides by 5 gives 0 < x.
Combining the solutions from both cases, we have x < 0 or 0 < x. This can be simplified to x ≠ 0.
C. 4x < -2x
Adding 2x to both sides gives 6x < 0. Dividing both sides by 6 gives x < 0.
D. -8x > 6x
Adding 8x to both sides gives 0 > 14x. Dividing both sides by 14 (and remembering to reverse the inequality because we're dividing by a negative value) gives 0 < x, or x > 0.
From the solutions obtained, we can see that the only true inequality is D. -8x > 6x.
To determine which of the inequalities is true, we need to solve each of them step-by-step. Let's go through each option one by one:
A. -3x > 5x
To solve this inequality, we can start by moving all terms with x to one side:
-3x - 5x > 0
Combine like terms:
-8x > 0
Now, we can divide both sides by -8. Remember that when you divide or multiply both sides of an inequality by a negative number, you need to flip the inequality sign.
x < 0
Therefore, the inequality -3x > 5x is not true.
B. |3x| < |2x|
For this inequality, we can consider two cases, one when x is positive, and the other when x is negative.
Case 1: x > 0
In this case, the absolute value of a positive number is itself.
3x < 2x
Subtract 2x from both sides:
x < 0
Case 2: x < 0
In this case, the absolute value of a negative number is its absolute value with the opposite sign.
-3x < 2x
Add 3x to both sides:
0 < 5x
Divide both sides by 5:
0 < x
Since x is less than 0 in Case 1 (x < 0) and greater than 0 in Case 2 (0 < x), we can say that this inequality is true for all real numbers.
C. 4x < -2x
Let's solve this inequality step-by-step:
Add 2x to both sides:
6x < 0
Divide both sides by 6:
x < 0
Therefore, the inequality 4x < -2x is true for x < 0.
D. -8x > 6x
Solve this inequality step-by-step:
Subtract 6x from both sides:
-14x > 0
Divide both sides by -14. Remember to flip the inequality sign when dividing by a negative number:
x < 0
Therefore, the inequality -8x > 6x is not true.
In conclusion, the inequality that is true is option B. |3x| < |2x|.