Question

In which quadrants do solutions for the inequality y ≤ (2/3)x - 4 exist?

1) i, iii, and iv
2) i, ii, and iii
3) i and iv
4) all four quadrants

Answer 1

To find the quadrants in which the solutions for the inequality

`y ≤ (2/3)x - 4` exist, we need to determine the regions where the inequality is true.

The regions satisfying the inequality are quadrant
`I`, where
`x` and
`y` are both positive and quadrant
`IV`, where
`x` is positive and
`y` is negative.

Step-by-step explanation:

To find the quadrants in which the solutions for the inequality

`y ≤ (2/3)x - 4` exist, we need to determine the regions where the inequality is true.

The easiest way to do this is to draw the line
`y = (2/3)x - 4`. This line has a slope of
`2/3`, which means it goes up
`2` units for every
`3` units it goes to the right. The y-intercept is
`-4`, which means the line passes through the point
`(0, -4)`.

If we pick a point on either side of the line and substitute its coordinates into the inequality, we can determine if it is true or false. For example, if we pick the point
`(0, 0)`, we get
`y ≤ (2/3)(0) - 4`, which simplifies to
`0 ≤ -4`. Since this is false, the region containing the origin (quadrant II) does not satisfy the inequality.

Using this same method, we can test other points and find that the regions satisfying the inequality are quadrant
`I`, where
`I` and
`y` are both positive, and quadrant
`IV`, where
`x` is positive and
`y` is negative. Therefore, the correct answer is option 3)
`i` and
`iv`.