Get Your Students Graphing Inequalities with These Top Worksheets ... - Free Printable

Since the user has set both `ExplainAnswer` and `TeachTopic` to false, I will proceed directly to solving the problems presented in the image. The image contains six distinct inequalities that need to be graphed on coordinate planes.

Here is the step-by-step solution for each problem:

Problem 7) $x < -5$
1. Identify the boundary line: The inequality involves only $x$. This means the boundary is a vertical line at $x = -5$.
2. Determine the line style: Since the symbol is strictly "less than" ($<$), not "less than or equal to," the line should be dashed (or broken).
3. Determine the shading: We want values of $x$ that are less than -5. On a number line, these are to the left. Therefore, we shade the entire region to the left of the vertical line.

Problem 8) $y \leq \frac{4}{3}x - 4$
1. Identify the slope and y-intercept: This is in slope-intercept form ($y = mx + b$). The y-intercept ($b$) is $-4$, and the slope ($m$) is $\frac{4}{3}$.
2. Plot the intercept: Start by placing a solid dot at $(0, -4)$ because the symbol includes "equal to" ($\leq$).
3. Use the slope: From $(0, -4)$, go up 4 units and right 3 units to find the next point $(3, 0)$. Place a solid dot there. Connect them with a solid line.
4. Determine the shading: The inequality is "less than or equal to." This means we shade the region *below* the line.

Problem 9) $3x - 2y < 10$
1. Find the x and y intercepts to draw the line:
* If $x = 0$: $-2y = 10 \rightarrow y = -5$. Point: $(0, -5)$.
* If $y = 0$: $3x = 10 \rightarrow x = \frac{10}{3}$ (approx 3.33). Point: $(3.33, 0)$.
2. Draw the line: Connect these points with a dashed line because the sign is strict ($<$).
3. Test a point: Let's test $(0,0)$ to see which side to shade.
* Substitute into equation: $3(0) - 2(0) < 10$
* $0 < 10$. This is true.
4. Shade: Since $(0,0)$ works, shade the side containing the origin (above/left of the line).

Problem 10) $5x - 3y \leq -15$
1. Find the intercepts:
* If $x = 0$: $-3y = -15 \rightarrow y = 5$. Point: $(0, 5)$.
* If $y = 0$: $5x = -15 \rightarrow x = -3$. Point: $(-3, 0)$.
2. Draw the line: Connect these points with a solid line because of the $\leq$ sign.
3. Test a point: Test $(0,0)$.
* $5(0) - 3(0) \leq -15$
* $0 \leq -15$. This is false.
4. Shade: Since $(0,0)$ does not work, shade the opposite side (the side away from the origin, above/right of the line).

Problem 11) $y \geq 4$
1. Identify the boundary line: This involves only $y$. It is a horizontal line at $y = 4$.
2. Line style: The symbol is $\geq$ (greater than or equal to), so use a solid line.
3. Shading: We want $y$ values greater than 4. Shade the region above the horizontal line.

Problem 12) $x - y > 2$
1. Find intercepts:
* If $x = 0$: $-y = 2 \rightarrow y = -2$. Point: $(0, -2)$.
* If $y = 0$: $x = 2$. Point: $(2, 0)$.
2. Draw the line: Connect points $(0, -2)$ and $(2, 0)$ with a dashed line (strict inequality $>$).
3. Test a point: Test $(0,0)$.
* $0 - 0 > 2$
* $0 > 2$. False.
4. Shade: Shade the side that does *not* contain the origin (below/right of the line).

Final Answer:
The solutions involve graphing the following inequalities on their respective grids:
7) A vertical dashed line at $x=-5$ with shading to the left.
8) A solid line passing through $(0,-4)$ and $(3,0)$ with shading below.
9) A dashed line passing through $(0,-5)$ and approx $(3.3,0)$ with shading above/left.
10) A solid line passing through $(0,5)$ and $(-3,0)$ with shading above/right.
11) A horizontal solid line at $y=4$ with shading above.
12) A dashed line passing through $(0,-2)$ and $(2,0)$ with shading below/right.