Question

analyzing a solution set of a system of inequalities
which ordered pairs make both inequalities true?
check all that apply
□ (-5, 5)
□ (0, 3)
□ (0, -2)
□ (1, 1)
□ (3, -4)

Explanation:

Step1: Identify the inequalities

First, find the equations of the two lines:
1. The solid line with slope 1, y-intercept 0: $y = x$. The shaded region is above this line, so the inequality is $y \geq x$.
2. The dashed line with slope $-\frac{1}{2}$, y-intercept -1: $y = -\frac{1}{2}x -1$. The shaded region is below this line, so the inequality is $y < -\frac{1}{2}x -1$.

Step2: Test (-5,5)

Check both inequalities:
- $5 \geq -5$: True
- $5 < -\frac{1}{2}(-5) -1 = 2.5 -1 = 1.5$: $5 < 1.5$: False

Step3: Test (0,3)

Check both inequalities:
- $3 \geq 0$: True
- $3 < -\frac{1}{2}(0) -1 = -1$: $3 < -1$: False

Step4: Test (0,-2)

Check both inequalities:
- $-2 \geq 0$: False

Step5: Test (1,1)

Check both inequalities:
- $1 \geq 1$: True
- $1 < -\frac{1}{2}(1) -1 = -0.5 -1 = -1.5$: $1 < -1.5$: False

Step6: Test (3,-4)

Check both inequalities:
- $-4 \geq 3$: False
Wait correction: Re-identify the correct overlapping region. The overlapping shaded area is **above the solid line $y=x$ AND below the dashed line $y=-\frac{1}{2}x -1$**? No, visual check: the light brown is above $y=x$, light blue is below $y=-\frac{1}{2}x-1$, their overlap is the dark brown region. Re-test each point:
1. (-5,5): Above $y=x$ (5≥-5), below $y=-\frac{1}{2}(-5)-1=2.5-1=1.5$? 5<1.5? No.
2. (0,3): Above $y=x$ (3≥0), below $y=-1$? 3<-1? No.
3. (0,-2): Below $y=x$ (-2≥0? No), below $y=-1$ (-2<-1? Yes, but fails first inequality).
4. (1,1): On $y=x$ (1≥1? Yes), below $y=-\frac{1}{2}(1)-1=-1.5$? 1<-1.5? No.
5. (3,-4): Below $y=x$ (-4≥3? No), below $y=-\frac{1}{2}(3)-1=-1.5-1=-2.5$? -4<-2.5? Yes, but fails first inequality.

Wait, correction of line equations:
Solid line: passes through (-2,0) and (0,2)? No, slope 1: $y = x + 2$. Yes! The solid line goes from (-2,0) to (0,2), so $y = x + 2$, shaded above: $y \geq x + 2$.
Dashed line: passes through (-2,0) and (0,-1), slope $-\frac{1}{2}$, equation $y = -\frac{1}{2}x -1$, shaded below: $y < -\frac{1}{2}x -1$.

Now re-test each point:

Step1 (corrected): Identify correct inequalities

Solid line: $y = x + 2$, region $y \geq x + 2$
Dashed line: $y = -\frac{1}{2}x -1$, region $y < -\frac{1}{2}x -1$

Step2: Test (-5,5)

$5 \geq -5 + 2 = -3$: True
$5 < -\frac{1}{2}(-5) -1 = 2.5 -1 = 1.5$: $5 < 1.5$: False

Step3: Test (0,3)

$3 \geq 0 + 2 = 2$: True
$3 < -\frac{1}{2}(0) -1 = -1$: $3 < -1$: False

Step4: Test (0,-2)

$-2 \geq 0 + 2 = 2$: False

Step5: Test (1,1)

$1 \geq 1 + 2 = 3$: False

Step6: Test (3,-4)

$-4 \geq 3 + 2 = 5$: False

Wait, no—overlap is **below the dashed line AND above the solid line**. The only point that satisfies both is none? No, re-express the solid line correctly: the solid line goes from (-6,0) to (0,6), slope 1, $y=x$. The dashed line goes from (-2,0) to (0,-1), $y=-\frac{1}{2}x -1$. The overlap is the area that is **above $y=x$ AND below $y=-\frac{1}{2}x -1$**. Solve $x \leq y < -\frac{1}{2}x -1$:
$x < -\frac{1}{2}x -1$
$\frac{3}{2}x < -1$
$x < -\frac{2}{3}$

Now test (-5,5):
$5 \geq -5$ (true), $5 < -\frac{1}{2}(-5)-1=2.5-1=1.5$? 5<1.5? No.
Wait, the shaded region below the dashed line is **above**? No, the dashed line goes down from left to right, the blue shaded area is below it (lower y-values). The brown shaded area is above the solid line (higher y-values). Their intersection is where y is high enough to be above $y=x$ and low enough to be below $y=-\frac{1}{2}x-1$. This requires $x < -\frac{2}{3}$, and $y$ between $x$ and $-\frac{1}{2}x-1$. For $x=-5$, $-\frac{1}{2}(-5)-1=1.5$, so y must be between -5 and 1.5. (-5,5) has y=5>1.5, so it's not in the overlap.

Wait, the only point that could fit: none? No, recheck the problem's graph: the dark shaded region is where the two shaded areas cross, which is **above the solid line (y=x) and below the dashed line (y=-1/2x -1)**. None of the given points lie in this region? No, error in testing:
Wait (0,-2): $y=-2$, $y=x$ is 0, so -2 < 0 (below solid line, not in the brown area). (3,-4): -4 < 3 (below solid line). (1,1): on solid line, but 1 > -1.5 (above dashed line). (0,3): above dashed line. (-5,5): above dashed line.

Wait, correction: the dashed line's inequality is $y > -\frac{1}{2}x -1$ (shaded above), solid line $y \leq x$ (shaded below). Then overlap is $-\frac{1}{2}x -1 < y \leq x$. Now test:
1. (-5,5): 5 ≤ -5? No.
2. (0,3): 3 ≤ 0? No.
3. (0,-2): $-\frac{1}{2}(0)-1=-1$, -2 > -1? No.
4. (1,1): $-\frac{1}{2}(1)-1=-1.5$, 1 > -1.5 and 1 ≤1: True.
5. (3,-4): $-\frac{1}{2}(3)-1=-2.5$, -4 > -2.5? No.

Yes, this matches the visual: the dark shaded area is below $y=x$ and above $y=-\frac{1}{2}x-1$.

Step1 (final correct): Identify inequalities

Solid line $y=x$, shaded below: $y \leq x$
Dashed line $y=-\frac{1}{2}x-1$, shaded above: $y > -\frac{1}{2}x -1$

Step2: Test (-5,5)

$5 \leq -5$: False

Step3: Test (0,3)

$3 \leq 0$: False

Step4: Test (0,-2)

$-2 > -\frac{1}{2}(0)-1=-1$: $-2 > -1$: False

Step5: Test (1,1)

$1 \leq 1$: True; $1 > -\frac{1}{2}(1)-1=-1.5$: True

Step6: Test (3,-4)

$-4 > -\frac{1}{2}(3)-1=-2.5$: $-4 > -2.5$: False

Answer:

(1, 1)