Question

Write a system of linear inequalities represented by the graph.

Inequality 1:
Inequality 2:
(Please explain, thanks!!)

Answer 1

• y ≥ x and
• y > - 2/3x + 2

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Find the lines first, then determine inequalities based on the shaded region and types of lines.

One line has positive slope and passes through the origin. Its x and y-coordinates are same, hence it is:

• y = x

The shaded region is above this line and the line is solid, therefore the inequality is:

• y ≥ x

The second line has a negative slope and it has y-intercept at (0, 2), x- intercept at (3, 0).

Its slope is m = (0 - 2)/(3 - 0) = - 2/3.

So the line is:

• y = - 2/3x + 2

Again, the shaded region is above the line, and the line is dashed, hence the inequality is:

• y > - 2/3x + 2

Answer 2

y > -2/3x + 2, y
`\geq` x

`\left \{ {{y > -2/3x + 2} \atop {y \geq x}} \right.`

Explanation:

Knowledge Needed

A system of linear inequalities is graphing 2 lines.

If the line's y side is less than it's x-side (y </
`\leq` x), shade below.

If the line's y side is greater than it's x-side (y </
`\leq` x), shade above.

If the line is dotted, it is < or >.

If the line is filled in, it is
`\leq` or
`\geq`.

Where both regions are shaded is the solution.

Question

Examine the dotted line.

1) We know it is < or > because it is dotted.

2) We know it is y > .... because if it was less than, it would be shaded below, which is impossible because the shaded region is above.

3) We know the y-intercept, when x=0, is 2.

4) We know the slope, how much the y-value of a line changed when the x-value changes by 1, is -2/3.

y > -2/3x + 2

Examine the solid line.

1) We know it is
`\geq` or
`\leq` because it is solid.

2) We know it is y
`\geq` .... because if it was less than, it would be shaded below, which is impossible because the shaded region is above.

3) We know the y-intercept, when x=0, is 0.

4) We know the slope, how much the y-value of a line changed when the x-value changes by 1, is 1.

y
`\geq` 1x + 0

y
`\geq` x

The solution to the linear inequality is when both lines are shaded in (true), which means both lines are the solution together.

y > -2/3x + 2, y
`\geq` x