To graph the inequality \( 5x - 7y < -35 \), you can follow these steps:
Step 1: Convert the inequality to an equation
First, convert the inequality to an equation by replacing the inequality sign with an equal sign: \[ 5x - 7y = -35 \]
Step 2: Find the intercepts
To graph the equation, you can find the x-intercept and y-intercept.
For the x-intercept (where \( y = 0 \)): \[ 5x - 7(0) = -35 \ 5x = -35 \ x = -7 \] So, the x-intercept is at the point \( (-7, 0) \).
For the y-intercept (where \( x = 0 \)): \[ 5(0) - 7y = -35 \ -7y = -35 \ y = 5 \] So, the y-intercept is at the point \( (0, 5) \).
Step 3: Plot the line
Draw a dashed line through the points \( (-7, 0) \) and \( (0, 5) \). The dashed line indicates that the points on the line are not included in the solution set (since the original inequality is \( < \)).
Step 4: Determine which side of the line to shade
To find out which side of the line to shade, pick a test point not on the line. A common choice is the origin \( (0, 0) \).
Substituting \( (0, 0) \) into the original inequality: \[ 5(0) - 7(0) < -35 \ 0 < -35 \quad (\text{This is false}) \] Since the test point \( (0, 0) \) does not satisfy the inequality, shade the opposite side of the line.
Step 5: Final graph
You should have a dashed line representing \( 5x - 7y = -35 \) and shading in the region that does not include the origin. This shaded area represents all the solutions to the inequality \( 5x - 7y < -35 \).
And that's how you graph the inequality!
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