Question

1. below are two inequalities and the graphs of their lines without the shading. by imagining where the shading should be, identify which point would satisfy both inequalities.

$y < -\frac{1}{3}x + 5$
$y > 6x - 1$
a. $(-4, -1)$
b. $(7, -8)$
c. $(3, 6)$
d. $(0, 7)$

1. below are two inequalities and the graphs of their lines without the shading. by imagining where the shading should be, identify which point would satisfy both inequalities.

$y < -6x - 1$
$y > \frac{1}{2}x + 2$
a. $(9, 3)$
b. $(1, 5)$
c. $(-6, 7)$
d. $(-2, -1)$

Explanation:

Problem 34

Step1: Test Point A (-4, -1)

For \( y < -\frac{1}{3}x + 5 \): Substitute \( x = -4 \), \( y = -1 \).
\( -1 < -\frac{1}{3}(-4) + 5 = \frac{4}{3} + 5 = \frac{19}{3} \approx 6.33 \). True.
For \( y > 6x - 1 \): Substitute \( x = -4 \), \( y = -1 \).
\( -1 > 6(-4) - 1 = -24 - 1 = -25 \). True.

Step2: Test Point B (7, -8)

For \( y < -\frac{1}{3}x + 5 \): \( -8 < -\frac{7}{3} + 5 = \frac{8}{3} \approx 2.67 \). True.
For \( y > 6x - 1 \): \( -8 > 42 - 1 = 41 \). False.

Step3: Test Point C (3, 6)

For \( y < -\frac{1}{3}x + 5 \): \( 6 < -1 + 5 = 4 \). False.

Step4: Test Point D (0, 7)

For \( y < -\frac{1}{3}(0) + 5 = 5 \): \( 7 < 5 \). False.

Step1: Test Point A (9, 3)

For \( y < -6x - 1 \): \( 3 < -54 - 1 = -55 \). False.

Step2: Test Point B (1, 5)

For \( y < -6(1) - 1 = -7 \): \( 5 < -7 \). False.

Step3: Test Point C (-6, 7)

For \( y < -6(-6) - 1 = 35 \): \( 7 < 35 \). True.
For \( y > \frac{1}{2}x + 2 \): \( 7 > \frac{1}{2}(-6) + 2 = -3 + 2 = -1 \). True.

Step4: Test Point D (-2, -1)

For \( y > \frac{1}{2}(-2) + 2 = -1 + 2 = 1 \): \( -1 > 1 \). False.

Answer:

A. (-4, -1)

Problem 35