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Graph y≤1−3x.

A. A graph with y axis ranging from negative 4 to 2 and x axis ranging from negative 1 to 4. A dotted line with arrows at both ends passing through the y axis at one and the x axis at point 3. The graph is shaded above the line.

B. Number graph x axis ranging from negative one to five, and y axis ranging from negative four to two. A solid line with arrows at both ends passing through the y axis at one and the x axis at point three. The graph is shaded below the line.

C. Number graph with y axis ranging from negative 4 to 2, and x axis ranging from negative 2 to 6. A solid line with arrows at both ends passing through the y axis at one and the x axis at point three. The graph is shaded above the line.

D. Number graph with x axis ranging from negative one to five, and y axis ranging from negative four to two. A dotted line with arrows at both ends passing through the y axis at one and the x axis at point three. The graph is shaded below the line.

3 answers

To graph the inequality \( y \leq 1 - 3x \), we first need to determine the characteristics of the line represented by the equation \( y = 1 - 3x \).

  1. Finding intercepts:

    • Y-intercept: Set \( x = 0 \): \[ y = 1 - 3(0) = 1 \quad \text{(point: (0, 1))} \]
    • X-intercept: Set \( y = 0 \): \[ 0 = 1 - 3x \implies 3x = 1 \implies x = \frac{1}{3} \quad \text{(point: } \left( \frac{1}{3}, 0 \right) \text{)} \]

  2. Graphing the line:

    • The line will pass through the points \( (0, 1) \) and \( \left( \frac{1}{3}, 0 \right) \).
    • This is a linear equation with a negative slope (-3), so the line will decrease as \( x \) increases.

  3. Line Style:

    • The inequality is \( y \leq 1 - 3x \), so the line should be solid (indicating points on the line are included), and the area below the line will be shaded (since we are considering values less than or equal to the line).

Based on this analysis, here are the options evaluated:

  • Option A: Dotted line (incorrect) and shaded above (incorrect).
  • Option B: Solid line (correct), but shaded below (incorrect).
  • Option C: Solid line (correct), but it says shaded above (incorrect).
  • Option D: Dotted line (incorrect) and shaded below (incorrect).

The correct graph should have a solid line passing through \( (0, 1) \) and \( ( \frac{1}{3}, 0 ) \), and it must be shaded below the line. Since none of the provided options meet all these criteria accurately, a perfect match isn't available in the choices given.

However, Option B has the correct line style but shades below, which is correct for the inequality. Therefore, the answer that is closest to being partially correct would be Option B, while noting the misunderstanding regarding shading.

Which ordered pairs are solutions to the inequality 2x+y>−4?
Select each correct answer.

A. (5, −12)

B. (−3, 0)

C. (0, 1)

D. (4, −12)

E. (−1, −1)

To determine which ordered pairs are solutions to the inequality \( 2x + y > -4 \), we can substitute each pair into the inequality and see if the resulting statement is true.

  1. Option A: \( (5, -12) \) \[ 2(5) + (-12) > -4 \ 10 - 12 > -4 \ -2 > -4 \quad \text{(True)} \]

  2. Option B: \( (-3, 0) \) \[ 2(-3) + 0 > -4 \ -6 > -4 \quad \text{(False)} \]

  3. Option C: \( (0, 1) \) \[ 2(0) + 1 > -4 \ 0 + 1 > -4 \ 1 > -4 \quad \text{(True)} \]

  4. Option D: \( (4, -12) \) \[ 2(4) + (-12) > -4 \ 8 - 12 > -4 \ -4 > -4 \quad \text{(False)} \]

  5. Option E: \( (-1, -1) \) \[ 2(-1) + (-1) > -4 \ -2 - 1 > -4 \ -3 > -4 \quad \text{(True)} \]

Summary of Results

  • Option A: True
  • Option B: False
  • Option C: True
  • Option D: False
  • Option E: True

Correct Solutions

The ordered pairs that are solutions to the inequality \( 2x + y > -4 \) are:

  • A. (5, −12)
  • C. (0, 1)
  • E. (−1, −1)