Linear Inequalities in Two Variables| Graphs & Equations ... - Free Printable

Let’s figure out which points lie in the yellow region.

The yellow region is above the red dashed line. The red dashed line goes through points like (0,2) and (4,0). Let’s find its equation to check each point.

Step 1: Find the slope of the red dashed line.
It passes through (0,2) and (4,0).
Slope = (change in y) / (change in x) = (0 - 2) / (4 - 0) = -2/4 = -1/2

Step 2: Write the equation of the line.
Using slope-intercept form: y = mx + b
We know m = -1/2 and when x=0, y=2 → so b=2.
Equation: y = (-1/2)x + 2

Step 3: The yellow region is ABOVE this line.
So for a point (x,y) to be in the yellow region, it must satisfy:
y > (-1/2)x + 2

But note: some points are ON the line — those are NOT in the yellow region because the line is dashed (meaning not included).

Now let’s test each labeled point:

1. (-1, 3)
Plug into inequality: Is 3 > (-1/2)(-1) + 2?
→ 3 > 0.5 + 2 → 3 > 2.5 → TRUE → Yellow region ✔

2. (0, 3)
Is 3 > (-1/2)(0) + 2? → 3 > 0 + 2 → 3 > 2 → TRUE → Yellow region ✔

3. (2, 3)
Is 3 > (-1/2)(2) + 2? → 3 > -1 + 2 → 3 > 1 → TRUE → Yellow region ✔

4. (1.5, 2)
Is 2 > (-1/2)(1.5) + 2? → 2 > -0.75 + 2 → 2 > 1.25 → TRUE → Yellow region ✔

5. (2, 1)
Is 1 > (-1/2)(2) + 2? → 1 > -1 + 2 → 1 > 1 → FALSE (equal, not greater) → NOT in yellow ✘

6. (3, 1)
Is 1 > (-1/2)(3) + 2? → 1 > -1.5 + 2 → 1 > 0.5 → TRUE → Yellow region ✔

Wait — hold on! Let me double-check (3,1):
(-1/2)*3 = -1.5; -1.5 + 2 = 0.5 → 1 > 0.5 → yes, true → should be in yellow.

But looking at the graph, (3,1) is below the red dashed line? Wait — let me recalculate the line value at x=3.

At x=3: y_line = (-1/2)(3) + 2 = -1.5 + 2 = 0.5
Point (3,1) has y=1, which is greater than 0.5 → so it IS above the line → should be in yellow.

But visually, in the image, (3,1) appears to be below the red dashed line? That can’t be right if our math says otherwise.

Wait — maybe I misread the graph. Let me check another point.

Check (4.5, 0):
y_line at x=4.5: (-1/2)(4.5) + 2 = -2.25 + 2 = -0.25
Point y=0 > -0.25 → TRUE → should be in yellow.

And indeed, (4.5,0) is shown as a blue dot in the yellow region.

Now check (2,1) again:
y_line at x=2: (-1/2)(2)+2 = -1+2=1 → point y=1 → equal → not greater → not in yellow → correct, it's on the line.

Now (3,1): y=1, line gives 0.5 → 1>0.5 → should be in yellow.

But in the image, (3,1) is drawn as a blue dot — and all blue dots seem to be in the yellow region? Let me list all points with their colors from the image description:

From the image text:

Pink/magenta dots: (-1,3), (2,1), (1.5,2) — wait no, (1.5,2) is magenta? But we calculated it’s in yellow.

Actually, let’s ignore color for now — focus on position relative to line.

List all points and whether they satisfy y > (-1/2)x + 2:

- (-1,3): 3 > 0.5 + 2 → 3>2.5 → YES
- (0,3): 3 > 0 + 2 → 3>2 → YES
- (2,3): 3 > -1 + 2 → 3>1 → YES
- (1.5,2): 2 > -0.75 + 2 → 2>1.25 → YES
- (2,1): 1 > -1 + 2 → 1>1 → NO (on line)
- (3,1): 1 > -1.5 + 2 → 1>0.5 → YES
- (0,0): 0 > 0 + 2 → 0>2 → NO
- (4.5,0): 0 > -2.25 + 2 → 0 > -0.25 → YES

So points in yellow region: (-1,3), (0,3), (2,3), (1.5,2), (3,1), (4.5,0)

Points NOT in yellow: (2,1) [on line], (0,0) [below]

But in the image, (2,1) is pink and on the line — correct.

(0,0) is labeled but not colored? Or is it white background? Anyway, not in yellow.

Now, the question is: “Which points lie in the yellow region?”

Based on calculation, the points that satisfy y > (-1/2)x + 2 are:

(-1,3), (0,3), (2,3), (1.5,2), (3,1), (4.5,0)

But let me verify (3,1) once more because visually it might look close.

At x=3, line y=0.5, point y=1 → definitely above.

Similarly, (4.5,0): line y=-0.25, point y=0 → above.

All good.

Final list of points in yellow region:

(-1, 3), (0, 3), (2, 3), (1.5, 2), (3, 1), (4.5, 0)

Note: (2,1) is on the boundary — not included since line is dashed.

(0,0) is below — not included.

So these six points are in the yellow region.

Final Answer:
(-1, 3), (0, 3), (2, 3), (1.5, 2), (3, 1), (4.5, 0)