To solve the system of inequalities \(y \geq x + 8\) and \(y \geq -2x - 7\) graphically, we can follow these steps:
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Graph the Lines: First, we will convert the inequalities to equations to identify the boundary lines.
- For the first inequality \(y = x + 8\), this is a straight line with a slope of \(1\) and a y-intercept of \(8\). The line will go through the points \((0, 8)\) and \((-8, 0)\).
- For the second inequality \(y = -2x - 7\), this line has a slope of \(-2\) and a y-intercept of \(-7\). The line will go through the points \((0, -7)\) and \((1, -9)\).
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Shading the Regions:
- For \(y \geq x + 8\), we shade the region above the line.
- For \(y \geq -2x - 7\), we shade the region above this second line as well.
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Finding the Intersection:
- To find the exact coordinates of the intersection of the two lines, we can set the equations equal to each other: \[ x + 8 = -2x - 7 \] Combine like terms: \[ x + 2x = -7 - 8 \] \[ 3x = -15 \quad \Rightarrow \quad x = -5 \] Now plug \(x = -5\) into one of the original equations to find \(y\): \[ y = -5 + 8 = 3 \] Thus, the intersection point is \((-5, 3)\).
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Overall Solution Set: The solution set comprises all points above both lines, including the lines themselves since the inequalities are "greater than or equal to".
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Point in the Solution Set: Any point that lies in the shaded region (above both lines) will be part of the solution set. A simple choice is the vertex we found:
The coordinates of a point in the solution set can be given as \((-5, 3)\).
Remember, when graphing, the shaded region is the solution space where both inequalities are satisfied.
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