Compare Lines with Slopes -3 and -6: Finding the Greater X-Axis Angle

Two lines haves slopes of -3 and -6.

Which of the lines forms a greater angle with the x axis?

To find which line forms a greater angle with the x-axis, we use the formula $\theta = \tan^{-1}(m)$. This gives us the angles formed by lines with slopes $m_1 = -3$ and $m_2 = -6$.

We first calculate the inverse tangent for both slopes:

• For $m_1 = -3$, the angle is $\theta_1 = \tan^{-1}(-3)$.
• For $m_2 = -6$, the angle is $\theta_2 = \tan^{-1}(-6)$.

The greater the absolute value of the negative slope, the closer the line is to the vertical, thus forming a greater angle with the x-axis.

Comparing $|-3|$ and $|-6|$, we see that $-6$ has a greater absolute value, indicating a steeper angle. Hence, although $-6$ is steeper, it forms a smaller angle with the x-axis because we're considering angles formed in the positive x-direction (i.e., above the x-axis).

Therefore, the line whose slope is $-3$ forms the greater angle with the x-axis.

The line whose slope is -3 forms the greater angle.