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The coordinate plane is a two-dimensional surface where we can visualize mathematical problems. It consists of a horizontal axis, called the x-axis, and a vertical axis, called the y-axis. These two axes intersect at the origin, which is the point (0,0). Every point on the coordinate plane can be identified by an ordered pair of numbers

• the first number representing the x-coordinate
• the second number representing the y-coordinate

This setup allows us to graph equations and inequalities by plotting points and lines. For instance, when graphing the inequality from part a, we plot the boundary line on this coordinate plane using specific coordinates. This geometric representation helps us better understand relationships between numbers and equations.

The slope-intercept form is a way of writing linear equations so that they reveal important characteristics of the line, such as the slope and y-intercept. It's expressed as:\[ y = mx + b \]Here, "m" represents the slope, and "b" is the y-intercept. With the inequality in part a given as:\[ y + 2x \geq 0 \]we transform it to the slope-intercept form to better understand it:\[ y \geq -2x \]From this, we see that the line has a slope of -2, meaning the line descends two units vertically for every unit it moves horizontally. The y-intercept is at the origin, (0,0), making it easier to start plotting the line on a coordinate plane. The slope-intercept form is particularly useful because it allows us to quickly identify these key features and translate them into a visual representation, aiding in the graphing process.

A boundary line is a crucial element when graphing linear inequalities. It represents the line where the inequality changes from true to false, or vice versa. In the inequality in part a,\[ y + 2x \geq 0 \]the boundary line is given by the equation\[ y = -2x \]This line is drawn solid because the inequality includes "equal to" (\geq\u001c), meaning points on the line satisfy the inequality. Plotting the boundary line involves using the slope to determine multiple points on the line, usually starting from the y-intercept. For part b, the inequality\[ y + 2x < 0 \]requires a dashed boundary line to show that points on the line do not satisfy the strict "less than" inequality.

Shading regions on the graph indicates all the solutions that satisfy an inequality. Once the boundary line is drawn, determine the side of the line that meets the inequality condition by using a test point or understanding the inequality's requirements. For the given inequality in part a,\[ y + 2x \geq 0 \]the shaded region is above the line\[ y = -2x \]because all points in that region have "y" values greater than or equal to\[ -2x \]In the second inequality\[ y + 2x < 0 \]we shade the area below the same line, as those points provide smaller "y" values than the equation dictates. Using shading effectively communicates which parts of the coordinate plane satisfy the inequality, visualizing many possible solutions as opposed to just points on the line itself.