Q. Graph this inequality: \(x – 3y > 6\).

Q: How do I rewrite x-3y>6 in slope-intercept form?

Solve for y: x-3y>6 \Rightarrow -3y>6-x. Divide by -3 and flip the inequality: y<\frac{x-6}{3}.

Q: Is the boundary line solid or dashed?

Because the inequality is strict (>), draw a dashed line for the boundary y=\frac{x-6}{3} to show points on the line are not included.

Q: Which side of the line do I shade?

Shade the half-plane where y<\frac{x-6}{3}, i.e., below the dashed line. You can confirm by testing a point like (0,0): 0<\frac{0-6}{3} is false, so (0,0) is not shaded.

Q: How do I test whether a point satisfies the inequality?

Substitute the point into x-3y>6. Example: (7,0): 7-3(0)=7>6 true, so (7,0) is included. Use easy points (0,0) or intercepts..

Q: What are the intercepts of the boundary line?

For y=0: x=6 gives (6,0). For x=0: -3y>6\Rightarrow y=-2 at equality, so (0,-2) lies on the boundary line.

Q: How can I list integer solutions quickly?

Choose integer x and compute y<\frac{x-6}{3}. Example x=9\Rightarrow y<1 so integer y\le 0. For x=6\Rightarrow y<0 so integer y\le -1..

Answer

Solve for \(y\).

\(x-3y>6\)

Subtract \(x\) from both sides.

\(-3y>6-x\)

Divide both sides by \(-3\). Since we divide by a negative number, reverse the inequality sign.

\(y<\frac{x-6}{3}\)

The boundary line is \(y=\frac{x-6}{3}\). Since the inequality is strict, the boundary line should be dashed.

The solution region is below the line.

The intercepts are \((0,-2)\) and \((6,0)\).

Detailed Explanation

Write the inequality and isolate y.Start with the given inequality:
\(x – 3y > 6\)

Subtract \(x\) from both sides:

\(-3y > 6 – x\)

Now divide both sides by \(-3\). Remember that dividing both sides of an inequality by a negative number reverses the inequality sign. Thus:

\(y < \dfrac{6 – x}{-3}\)

Simplify the right-hand side:

\(y < \dfrac{6}{-3} + \dfrac{-x}{-3}\)

\(y < -2 + \dfrac{x}{3}\)

Reorder terms to the standard slope-intercept form:

\(y < \dfrac{1}{3}x – 2\)
Identify the boundary line and its characteristics.The boundary is the line obtained by replacing the inequality with equality:
\(y = \dfrac{1}{3}x – 2\)

From this form:
- The slope is \(\dfrac{1}{3}\).
- The y-intercept is \((0, -2)\).
- Because the original inequality is strict (\(>\) became \(<\) when solving for \(y\)), use a dashed line to indicate points on the line are not included in the solution.
Plot two points on the boundary line.Start with the y-intercept point:
\((0, -2)\)

Use the slope \(\dfrac{1}{3}\) which means “rise 1, run 3”. From \((0, -2)\) move right 3 and up 1 to get the second point:

\((3, -1)\)

Plot these two points and draw a dashed line through them to represent \(y = \dfrac{1}{3}x – 2\).
Determine which side of the line to shade.Pick a test point that is not on the boundary. A convenient choice is \((0, 0)\).
Substitute \((0,0)\) into the inequality \(y < \dfrac{1}{3}x – 2\):

Left side: \(y = 0\). Right side: \(\dfrac{1}{3}\cdot 0 – 2 = -2\).

Check \(0 < -2\). This is false, so \((0,0)\) is not in the solution region.

Because the test point above the line is not in the solution, the solution region is the opposite side of the line: shade the region below the dashed line \(y = \dfrac{1}{3}x – 2\).
Final description of the graph.Graph the dashed line \(y = \dfrac{1}{3}x – 2\) (dashed because the inequality is strict). Shade the region below this line. All points in that shaded region satisfy the original inequality \(x – 3y > 6\).

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Algebra FAQs

How do I rewrite \(x-3y>6\) in slope-intercept form?

Solve for \(y\): \(x-3y>6 \Rightarrow -3y>6-x.\) Divide by \(-3\) and flip the inequality: \(y<\frac{x-6}{3}\).

Is the boundary line solid or dashed?

Because the inequality is strict (>), draw a dashed line for the boundary \(y=\frac{x-6}{3}\) to show points on the line are not included.

Which side of the line do I shade?

Shade the half-plane where \(y<\frac{x-6}{3}\), i.e., below the dashed line. You can confirm by testing a point like \((0,0)\): \(0<\frac{0-6}{3}\) is false, so \((0,0)\) is not shaded.

How do I test whether a point satisfies the inequality?

Substitute the point into \(x-3y>6\). Example: \((7,0): 7-3(0)=7>6\) true, so \((7,0)\) is included. Use easy points \((0,0)\) or intercepts..

What are the intercepts of the boundary line?

For \(y=0\): \(x=6\) gives \((6,0)\). For \(x=0\): \(-3y>6\Rightarrow y=-2\) at equality, so \((0,-2)\) lies on the boundary line.

What is the slope of the boundary line and how do I graph it?

Do I need to flip the inequality sign at any step?

Yes: when dividing or multiplying both sides by a negative number (here -3) you must flip ">" to "<", giving \(y<\frac{x-6}{3}\)..

How can I list integer solutions quickly?

Choose integer \(x\) and compute \(y<\frac{x-6}{3}\). Example \(x=9\Rightarrow y<1\) so integer \(y\le 0\). For \(x=6\Rightarrow y<0\) so integer \(y\le -1\)..

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