Which graph represents the inequality 3x + y ≤ 1?
Understand the Problem
The question asks to identify the graph that correctly represents the given linear inequality. This involves understanding how to graph linear inequalities, including the solid or dashed line indicating inclusion or exclusion of the boundary, and the shaded region representing the solution set.
Answer
The answer is the correctly graphed linear inequality, showing the boundary line (solid or dashed) and shaded region.
Answer for screen readers
The solution will be the graph showing the correct boundary line (solid or dashed) and the correct shaded region.
Steps to Solve
- Convert the inequality to slope-intercept form (if necessary)
The slope-intercept form of a linear equation is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. This form makes it easy to graph the line. If your inequality is not already in this form, rearrange it to isolate $y$ on one side. Remember that if you multiply or divide by a negative number, you must flip the direction of the inequality sign.
- Graph the boundary line
Treat the inequality as an equality and graph the corresponding line. For example, if the inequality is $y > 2x + 1$, graph the line $y = 2x + 1$.
- Determine if the line is solid or dashed
If the inequality is strict (i.e., $<$ or $>$) use a dashed line to indicate that the points on the line are not included in the solution. If the inequality is non-strict (i.e., $\leq$ or $\geq$), use a solid line to indicate that the points on the line are included in the solution.
- Choose a test point
Pick a point that is not on the line. The easiest point to use is often the origin, $(0, 0)$, if the line doesn't pass through it.
- Substitute the test point into the inequality
Plug the $x$ and $y$ coordinates of the test point into the original inequality.
- Determine which side to shade
If the test point satisfies the inequality, shade the side of the line that contains the test point. If the test point does not satisfy the inequality, shade the side of the line that does not contain the test point. The shaded region represents all the solutions to the inequality.
The solution will be the graph showing the correct boundary line (solid or dashed) and the correct shaded region.
More Information
Keep in mind that some graphing tools might not perfectly display dashed lines. Look for subtle differences to distinguish between solid and dashed lines.
Tips
A common mistake is forgetting to flip the inequality sign when multiplying or dividing by a negative number. It's also easy to get the shading wrong, so always use a test point to check your answer. Another error occurs when you use the wrong type of line (dashed instead of solid or vice versa) to represent strict or non-strict inequalities.
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