For today's exit ticket, Mr. Samuels wants students to graph the inequality y ≥ -1/3x - 4 on the grid below.

Question image

Understand the Problem

The question is asking to graph the inequality y ≥ - rac{1}{3}x - 4 on a provided coordinate plane. The solution will involve identifying the boundary line and determining the shading for the inequality.

Answer

The graph includes a solid line for $y = -\frac{1}{3}x - 4$ with shading above the line for the inequality $y \geq -\frac{1}{3}x - 4$.
Answer for screen readers

The graph of the inequality $y \geq -\frac{1}{3}x - 4$ consists of a solid line representing the boundary $y = -\frac{1}{3}x - 4$ and the area above this line shaded.

Steps to Solve

  1. Identify the boundary line

To graph the inequality, we first need to graph the boundary line of the equation. The equation of the line is given by

$$ y = -\frac{1}{3}x - 4 $$

  1. Graph the boundary line

Since we have a "greater than or equal to" ($\geq$) inequality, we will graph the boundary line as a solid line. This indicates that points on the line are included in the solution set.

To plot the line, find two points:

  • Find the y-intercept: Set $x = 0$.

$$ y = -\frac{1}{3}(0) - 4 = -4 $$

This gives the point (0, -4).

  • Find another point: Set $x = 3$.

$$ y = -\frac{1}{3}(3) - 4 = -1 - 4 = -5 $$

This gives the point (3, -5).

Now, plot the points (0, -4) and (3, -5) on the coordinate plane and draw a solid line through them.

  1. Determine the shading for the inequality

Since the inequality is $y \geq -\frac{1}{3}x - 4$, we need to shade the region above the line because the values of $y$ that satisfy the inequality are greater than or equal to those on the line.

  1. Finalizing the graph

Make sure the area above the line is shaded, as this represents all the solutions to the inequality.

The graph of the inequality $y \geq -\frac{1}{3}x - 4$ consists of a solid line representing the boundary $y = -\frac{1}{3}x - 4$ and the area above this line shaded.

More Information

Inequalities can be graphed similarly to equations, but the key is to determine whether the line is solid or dashed. A solid line indicates that points on the line are included in the solution, whereas a dashed line would indicate they are not.

Tips

  • Forgetting to use a solid line for "greater than or equal to" ($\geq$).
  • Confusing the direction of the shading; always shade above the line for $\geq$.
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