To solve the system of inequalities by graphing, follow these steps:
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Rewrite the inequalities: We will rewrite the inequalities in slope-intercept form (y = mx + b), which makes them easier to graph.
First inequality:
\(3x + 4y < -14\)Solving for y: \[ 4y < -3x - 14 \] \[ y < -\frac{3}{4}x - \frac{14}{4} \] \[ y < -\frac{3}{4}x - \frac{7}{2} \]
Second inequality:
\(x - 3y \geq 17\)Solving for y: \[ -3y \geq -x + 17 \] \[ y \leq \frac{1}{3}x - \frac{17}{3} \]
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Graph the inequalities:
Inequality 1: \(y < -\frac{3}{4}x - \frac{7}{2}\)
- The boundary line is \(y = -\frac{3}{4}x - \frac{7}{2}\). This line will be dashed because the inequality is strict ("less than").
- Find two points to graph the line. For example:
- If \(x = 0\), then \(y = -\frac{7}{2} = -3.5\).
- If \(x = 4\), then \(y = -\frac{3}{4}(4) - \frac{7}{2} = -3 - 3.5 = -6.5\).
- Plot these points (0, -3.5) and (4, -6.5), and draw a dashed line through them. Shade the region below this line.
Inequality 2: \(y \leq \frac{1}{3}x - \frac{17}{3}\)
- The boundary line is \(y = \frac{1}{3}x - \frac{17}{3}\). This line will be solid because the inequality is inclusive ("greater than or equal to").
- Find two points to graph the line. For example:
- If \(x = 0\), then \(y = -\frac{17}{3} \approx -5.67\).
- If \(x = 6\), then \(y = \frac{1}{3}(6) - \frac{17}{3} = 2 - \frac{17}{3} = 2 - 5.67 = -3.67\).
- Plot these points (0, -5.67) and (6, -3.67), and draw a solid line through them. Shade the region below this line.
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Identify the solution region:
- The solution to the system of inequalities is the intersection of the shaded areas from both inequalities. This is the region where both conditions are met.
Summary
- The first inequality's region will be below the dashed line \(y = -\frac{3}{4}x - \frac{7}{2}\).
- The second inequality's region will be below the solid line \(y = \frac{1}{3}x - \frac{17}{3}\).
Make sure to check a few points in the overlapping shaded region to confirm they satisfy both inequalities.