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Use the image to answer the question.

Four points and two intersecting lines are plotted on a coordinate plane. The x and y axes range from negative 5 to 10 in unit increments, labeled at every 5 units.

Which of the points on the graph is a solution for the simultaneous inequalities y>4x−2
and y≥−1/3x+2
? Enter the x-value followed by the y-value.

(1 point)
(
,
)
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3 answers

To determine which point is a solution for the simultaneous inequalities \( y > 4x - 2 \) and \( y \geq -\frac{1}{3}x + 2 \), we need to analyze both inequalities.

  1. Graph the inequalities:

    • For \( y = 4x - 2 \):

      • This is a line with a slope of 4 and a y-intercept of -2. The area above this line will be where \( y > 4x - 2 \).

    • For \( y = -\frac{1}{3}x + 2 \):

      • This is a line with a slope of -1/3 and a y-intercept of 2. The area above (or on) this line will be where \( y \geq -\frac{1}{3}x + 2 \).

  2. Identify candidate points:

    • Look at the points plotted on the graph and check which points satisfy both inequalities.

  3. Testing Points:

    • For each candidate point, substitute the \( x \) value into both inequalities to check for validity.

Since I can't see the image to directly assess the points, please substitute any points you see on the graph into the inequalities to see if they hold true.

For example:

  • If a point is (1, 5): \[ y = 5 > 4(1) - 2 \rightarrow 5 > 2 \quad \text{(True)} \] \[ y = 5 \geq -\frac{1}{3}(1) + 2 \rightarrow 5 \geq \frac{5}{3} \quad \text{(True)} \]

If both conditions hold true for that point, it is a solution to the inequalities.

After examining the graph in this way, provide the coordinates of the point you found to be a solution.

If you have specific points or coordinates in the graph to check, please list them, and I can help you determine which one satisfies the inequalities.

the points are the graph are (-1, 5), (0.923, 1.6), ( 5, 2), (-4, -2)

Let's check each of the given points against the inequalities \( y > 4x - 2 \) and \( y \geq -\frac{1}{3}x + 2 \).

  1. Point (-1, 5):

    • For \( y > 4x - 2 \): \[ 5 > 4(-1) - 2 \rightarrow 5 > -4 - 2 \rightarrow 5 > -6 \quad \text{(True)} \]
    • For \( y \geq -\frac{1}{3}x + 2 \): \[ 5 \geq -\frac{1}{3}(-1) + 2 \rightarrow 5 \geq \frac{1}{3} + 2 \rightarrow 5 \geq \frac{1}{3} + \frac{6}{3} \rightarrow 5 \geq \frac{7}{3} \quad \text{(True)} \]
      • Thus, (-1, 5) satisfies both inequalities.

  2. Point (0.923, 1.6):

    • For \( y > 4x - 2 \): \[ 1.6 > 4(0.923) - 2 \rightarrow 1.6 > 3.692 - 2 \rightarrow 1.6 > 1.692 \quad \text{(False)} \]
    • This point does not satisfy the first inequality.

  3. Point (5, 2):

    • For \( y > 4x - 2 \): \[ 2 > 4(5) - 2 \rightarrow 2 > 20 - 2 \rightarrow 2 > 18 \quad \text{(False)} \]
    • This point does not satisfy the first inequality.

  4. Point (-4, -2):

    • For \( y > 4x - 2 \): \[ -2 > 4(-4) - 2 \rightarrow -2 > -16 - 2 \rightarrow -2 > -18 \quad \text{(True)} \]
    • For \( y \geq -\frac{1}{3}x + 2 \): \[ -2 \geq -\frac{1}{3}(-4) + 2 \rightarrow -2 \geq \frac{4}{3} + 2 \rightarrow -2 \geq \frac{4}{3} + \frac{6}{3} \rightarrow -2 \geq \frac{10}{3} \quad \text{(False)} \]
    • This point does not satisfy the second inequality.

Conclusion:

The only point that satisfies both inequalities \( y > 4x - 2 \) and \( y \geq -\frac{1}{3}x + 2 \) is: \[ \boxed{(-1, 5)} \]