Questions: Yes or No? (a) If x(x+3) > 0, does it follow that x is positive? Yes No If No, give an example. (If Yes, enter Yes.)

Transcript text: Yes or No? (a) If $x(x+3)>0$, does it follow that $x$ is positive? Yes No If No, give an example. (If Yes, enter Yes.)

Solution

Solution Steps

To determine if $ x(x+3) > 0 $ implies that $ x $ is positive, we need to analyze the inequality. The product $ x(x+3) $ is positive if both factors are either both positive or both negative. We will check the intervals where this condition holds.

Step 1: Understand the Problem

We need to determine if the inequality $ x(x+3) > 0 $ implies that $ x $ is positive. If the answer is "No," we should provide an example where $ x $ is not positive but the inequality still holds.

Step 2: Analyze the Inequality

The inequality $ x(x+3) > 0 $ can be analyzed by considering the product of two factors:

$ x $
$ x + 3 $

For the product $ x(x+3) $ to be greater than zero, both factors must be either both positive or both negative.

Step 3: Determine the Sign of Each Factor

$ x $ is positive if $ x > 0 $.
$ x + 3 $ is positive if $ x > -3 $.

Step 4: Consider the Intervals

We need to consider the intervals where the product $ x(x+3) $ is positive:

Both factors are positive: $ x > 0 $ and $ x + 3 > 0 $ which simplifies to $ x > 0 $.
Both factors are negative: $ x < 0 $ and $ x + 3 < 0 $ which simplifies to $ x < -3 $.

Step 5: Combine the Intervals

The inequality $ x(x+3) > 0 $ holds in the intervals:

$ x > 0 $
$ x < -3 $

Step 6: Conclusion

From the intervals, we see that $ x $ can be either greater than 0 or less than -3. Therefore, $ x $ does not have to be positive for the inequality to hold.

Step 7: Provide an Example

An example where $ x $ is not positive but the inequality $ x(x+3) > 0 $ holds is $ x = -4 $: \[ x = -4 \implies (-4)(-4+3) = (-4)(-1) = 4 > 0 \]