Graph this inequality: 5x - y < 6. Plot points on the boundary line. Select the region to shade it.

Understand the Problem

The question is asking us to graph the linear inequality 5x - y < 6. We need to identify the boundary line and then determine which region to shade according to the inequality.

Answer

The graph of the inequality $ 5x - y < 6 $ features a dashed line along $ y = 5x - 6 $ with the region above the line shaded.

Steps to Solve

Rewrite the inequality in slope-intercept form

Start by isolating ( y ) in the inequality ( 5x - y < 6 ):

$$ -y < -5x + 6 $$

Now, multiply by -1 (remember to flip the inequality):

$$ y > 5x - 6 $$

Identify the boundary line

The boundary line for the inequality can be found by replacing the inequality symbol with an equals sign:

$$ y = 5x - 6 $$

Plot points for the boundary line

Choose a couple of ( x ) values to find corresponding ( y ) values:

For ( x = 0 ):

$$ y = 5(0) - 6 = -6 $$

Point: ( (0, -6) )
For ( x = 2 ):

$$ y = 5(2) - 6 = 4 $$

Point: ( (2, 4) )

Draw the boundary line

Plot the points ( (0, -6) ) and ( (2, 4) ) on a graph and connect them with a dashed line since the inequality is strict (( > )).

Select the region to shade

To determine which side of the line to shade, pick a test point not on the line (commonly ( (0, 0) )). Substitute into the original inequality:

$$ 5(0) - 0 < 6 $$

This simplifies to ( 0 < 6 ), which is true. Therefore, shade the region that includes the point ( (0, 0) ).

The graph of the inequality ( 5x - y < 6 ) shows a dashed line for the boundary ( y = 5x - 6 ) with the region above the line shaded.

More Information

This inequality represents a half-plane above the line, indicating all combinations of ( x ) and ( y ) values that satisfy the condition ( 5x - y < 6 ). The use of a dashed line signifies that points on the line itself are not included in the solution set.

Tips

Forgetting to flip the inequality sign when multiplying or dividing by a negative number.
Not using a dashed line for strict inequalities (i.e., ( < ) or ( > )).
Shading the wrong region by not testing the correct point.

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