Which inequality is graphed on the coordinate plane, given a dotted line intersecting the x-axis at (-0.5, 0) and the y-axis at (0, 2), with the region to the left of the line shad... Which inequality is graphed on the coordinate plane, given a dotted line intersecting the x-axis at (-0.5, 0) and the y-axis at (0, 2), with the region to the left of the line shaded?
Understand the Problem
The question asks to identify the inequality represented by a given graph on the coordinate plane. The graph has a dotted line intersecting the x-axis at (-0.5, 0) and the y-axis at (0, 2). The region to the left of the line is shaded, indicating the solution set of the inequality. We need to determine which of the provided inequalities matches this description.
Answer
$4x - y < -2$
Answer for screen readers
$4x - y < -2$
Steps to Solve
- Find the equation of the line
The line passes through the points $(-0.5, 0)$ and $(0, 2)$. We can find the slope ($m$) using the formula:
$m = \frac{y_2 - y_1}{x_2 - x_1}$
Substituting the given points:
$m = \frac{2 - 0}{0 - (-0.5)} = \frac{2}{0.5} = 4$
- Write the equation of the line in slope-intercept form
The slope-intercept form is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. We know $m = 4$ and the y-intercept is $2$. So the equation of the line is:
$y = 4x + 2$
- Determine the inequality symbol
Since the line is dotted, the inequality will be either $>$ or $<$. The region to the left of the line is shaded. Let's test a point in the shaded region, such as $(-1, 0)$. Plug this point into the equation $y = 4x + 2$. $0 \ ? \ 4(-1) + 2$ $0 \ ? \ -4 + 2$ $0 \ ? \ -2$
Since $0 > -2$, the inequality is $y > 4x + 2$.
- Rewrite the inequality in the required format. The inequality can be rewritten as: $y > 4x + 2$
Rearranging the terms, we get: $y - 4x > 2$
Multiplying by -1 and flipping the inequality sign
$4x - y < -2$
$4x - y < -2$
More Information
The graphed inequality represents all the points $(x, y)$ in the coordinate plane that satisfy the condition $4x - y < -2$. The dotted line indicates that the points on the line itself are not included in the solution set. The shading to the left of the line visually represents all the points that satisfy the inequality.
Tips
- Using the wrong inequality symbol: Forgetting to flip the inequality sign when multiplying or dividing both sides by a negative number is a common mistake. Also, not considering whether the line is dashed or solid when determining whether to use $<$ or $\leq$ (or $>$ or $\geq$).
- Incorrectly calculating the slope: Mistakes in calculating the slope can lead to an incorrect equation for the line and, consequently, an incorrect inequality. Ensure the slope formula is applied correctly.
- Shading the wrong region: Choosing the wrong side of the line to shade will result in an incorrect inequality. Always test a point in the shaded region to verify the inequality.
- Algebraic manipulation errors: When rearranging the inequality, errors in algebraic manipulation can occur, such as incorrectly adding or subtracting terms, leading to a wrong final inequality.
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