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Four points and two intersecting lines are plotted on a coordinate plane. The x-axis ranges from negative 15 to 15 in unit increments, labeled at every 5. The y-axis ranges from negative 10 to 10 in unit increments, labeled at every 5.

Which of the points on the graph is a solution to the simultaneous inequalities y≤14x+2
and y>12x+3
? Enter the x-vavalue followed by the y-value.

(1 point)
(
,
)
Remaining Attempts : 3

1 answer

To determine which of the points on the graph satisfies the simultaneous inequalities \(y \leq 14x + 2\) and \(y > 12x + 3\), we need to analyze the inequalities.

  1. Inequality 1: \(y \leq 14x + 2\)

    • This line (14x + 2) represents all points below and on the line.

  2. Inequality 2: \(y > 12x + 3\)

    • This line (12x + 3) indicates that we are looking for points above this line.

To find points that satisfy both inequalities, they must lie in the region where:

  • They are below or on the line \(y = 14x + 2\).
  • They are above the line \(y = 12x + 3\).

Finding Points

Now, let's examine specific points \((x, y)\) one by one:

  1. Point 1 (for example, \((-5, -5)\):

    • Calculate for \(y = 14(-5) + 2 = -70 + 2 = -68\) (not satisfied)
    • Calculate for \(y = 12(-5) + 3 = -60 + 3 = -57\) (not satisfied)

  2. Point 2 (for example, \((0, 0)\):

    • Calculate for \(y = 14(0) + 2 = 2\) (satisfied)
    • Calculate for \(y = 12(0) + 3 = 3\) (not satisfied)

  3. Continue with other relevant points on the graph until one satisfies both inequalities.

Example Calculation

  1. Check point \((1, 5)\):

    • For \(y \leq 14x + 2 \implies 5 \leq 14(1) + 2 \implies 5 \leq 16\) (satisfied).
    • For \(y > 12x + 3 \implies 5 > 12(1) + 3 \implies 5 > 15\) (not satisfied).

  2. Check point \((1, 13)\):

    • For \(y \leq 14x + 2 \implies 13 \leq 14(1) + 2 \implies 13 \leq 16\) (satisfied).
    • For \(y > 12x + 3 \implies 13 > 12(1) + 3 \implies 13 > 15\) (not satisfied).

  3. Check point \((2, 10)\):

    • For \(y \leq 14x + 2 \implies 10 \leq 14(2) + 2 \implies 10 \leq 30\) (satisfied).
    • For \(y > 12x + 3 \implies 10 > 12(2) + 3 \implies 10 > 27\) (not satisfied).

Solution

Let's simplify and identify a point that satisfies both.

After further checking, you could denote a point through systematic checking until you locate \((x, y)\) aligned to the conditions described:

The point that works after thorough checking could potentially be something like: \[ (x, y) = (x_0, y_0) \text{ satisfying both conditions.} \] Please replace \((x, y)\) here with your validated final coordinates based on which points you've verified satisfy both inequalities.

If necessary, please share specific points available on your graph for a targeted assessment!