Solve the given equations geometrically, that is, interpreting the absolute value $|a-b|$ as the distance between $a$ and $b$.
(a) $|x-5|=2$
(c) $|3 x-7|=9$
(b) $|x+3|=1$
(d) $|x+2|=|x-5|$
(a) $|x-5|=2$ means that the distance of $x$ from 5 is 2. Therefore, the roots are 3 and 7.

![](https://cdn.mathpix.com/cropped/2024_05_01_30c9a294a58a6e4b8190g-061.jpg?height=128&width=659&top_left_y=1986&top_left_x=824)

(b) $|x+3|=1$ means that the distance of $x$ from -3 is 1. Therefore, the roots are -4 and -2.

![](https://cdn.mathpix.com/cropped/2024_05_01_30c9a294a58a6e4b8190g-061.jpg?height=123&width=659&top_left_y=2206&top_left_x=824)

(c) Denoting $y=3x$, the equation takes the form $|y-7|=9$, which means that the distance of $y$ from 7 is 9. Therefore, the roots are -2 and 16. Since $y=3x$, we have $3x=-2$ and $3x=16$, so the roots of the original equation are $x=-\frac{2}{3}$ and $x=\frac{16}{3}$.

![](https://cdn.mathpix.com/cropped/2024_05_01_30c9a294a58a6e4b8190g-061.jpg?height=120&width=659&top_left_y=2576&top_left_x=824)
(d) The roots of the equation $|x+2|=|x-5|$ are the numbers equidistant from -2 and 5. However, there can only be one number equidistant from two others, and it lies halfway between them. Since the distance from 5 to -2 is 7, the equidistant point must be 3.5 units away from -2 and 5. Therefore, the solution is $x=1.5$.

![](https://cdn.mathpix.com/cropped/2024_05_01_30c9a294a58a6e4b8190g-062.jpg?height=194&width=662&top_left_y=480&top_left_x=777)