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.)

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.

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.

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

1. $x$
2. $x + 3$

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

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

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

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

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

1. $x > 0$
2. $x < -3$

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.

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

Final Answer