1. **State the problem:** We need to find the coordinates of a point whose abscissa (x-coordinate) is $-4$ and which is at a distance of 15 units from the point $(5,-9)$. 2. **Formula used:** The distance $d$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ 3. **Apply the formula:** Let the point we want to find be $(-4, y)$. The distance from $(5,-9)$ to $(-4,y)$ is 15, so: $$15 = \sqrt{(-4 - 5)^2 + (y - (-9))^2}$$ 4. **Simplify inside the square root:** $$15 = \sqrt{(-9)^2 + (y + 9)^2}$$ $$15 = \sqrt{81 + (y + 9)^2}$$ 5. **Square both sides to eliminate the square root:** $$15^2 = 81 + (y + 9)^2$$ $$225 = 81 + (y + 9)^2$$ 6. **Isolate the squared term:** $$(y + 9)^2 = 225 - 81$$ $$(y + 9)^2 = 144$$ 7. **Take the square root of both sides:** $$y + 9 = \pm 12$$ 8. **Solve for $y$:** - If $y + 9 = 12$, then $y = 3$ - If $y + 9 = -12$, then $y = -21$ 9. **Final answer:** The points are $(-4, 3)$ and $(-4, -21)$. These points have an abscissa of $-4$ and are exactly 15 units away from $(5,-9)$.