To graph the inequality \( 2x - 7y < -14 \), follow these steps:
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Rewrite the inequality: First, it's often helpful to express the inequality in slope-intercept form (i.e., \( y = mx + b \)). Start by isolating \( y \): \[ 2x - 7y < -14 \] Subtract \( 2x \) from both sides: \[ -7y < -2x - 14 \] Now, divide every term by \(-7\). Remember that dividing or multiplying by a negative number reverses the inequality sign: \[ y > \frac{2}{7}x + 2 \]
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Graph the boundary line: The boundary line is given by the equation \( y = \frac{2}{7}x + 2 \). To graph this line, you can find two points:
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Y-intercept: Set \( x = 0 \): \[ y = \frac{2}{7}(0) + 2 = 2 \] This gives you the point \( (0, 2) \).
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X-intercept: Set \( y = 0 \): \[ 0 = \frac{2}{7}x + 2 \implies \frac{2}{7}x = -2 \implies x = -7 \] This gives you the point \( (-7, 0) \).
Plot these points and draw a line through them. Since the inequality is \( y > \frac{2}{7}x + 2 \) and not \( y \geq \frac{2}{7}x + 2 \), the line should be dashed to indicate that points on the line are not included in the solution.
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Test a point: To determine which side of the line to shade, choose a test point not on the line. A common choice is the origin \( (0, 0) \). Substitute \( (0, 0) \) into the original inequality: \[ 2(0) - 7(0) < -14 \implies 0 < -14 \] This statement is false, so we do not shade the side of the line that includes the origin.
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Shade the correct area: Since the inequality is \( y > \frac{2}{7}x + 2 \), you would shade the area above the dashed line.
Your final graph should show a dashed line representing \( y = \frac{2}{7}x + 2 \) with the area above the line shaded.