Question

Check whether \((0,0)\) is a solution. Then sketch the graph of the inequality. $$ y<-2 $$

Short answer

(0,0) is not a solution. The graph of this inequality will be a dashed horizontal line y = -2, and all area below this line will be shaded.

Step-by-step solution

Verify the solution

To verify the solution, we substitute x=0, y=0 into the inequality \(y < -2\). We get '0 < -2'. This is false, meaning (0,0) is not a solution to the inequality.

Sketch the graph

In plotting the graph of \(y < -2\), the y-coordinate where the horizontal line should cross is at -2. The line needs to be dashed (not solid), as 'less than (-2)' does not include -2 itself. Then, the area below the line should be shaded to represent the region where the condition \(y < -2\) is satisfied. This represents all of the solutions to the inequality.

Key concepts

Inequality Solutions

Understanding inequality solutions is essential for solving many mathematical problems, particularly when we deal with ranges of values rather than fixed numbers. An inequality like \(y < -2\) indicates a relation in which one side of the inequality is less than the other side. In our case, the inequality states that the 'y' value must be less than -2. When seeking solutions to such inequalities, we’re often looking for all the possible 'y' values that satisfy this condition. Unlike equations, which usually have a fixed solution, inequalities represent a set of possible solutions, often visualized as a continuous range.

• Always replace the variable with numbers to test if an inequality holds true.
• Use a number line to visually assess the range of solutions for one-variable inequalities.
• Remember, solutions to inequalities are often in ranges or regions, not just specific points.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface defined by two number lines that intersect at right angles: the x-axis (horizontal) and the y-axis (vertical). The plane is divided into four quadrants, each with a unique combination of positive and negative values for 'x' and 'y'. Points on this plane are given as ordered pairs (x, y), where 'x' represents the position along the horizontal axis and 'y' signifies the position along the vertical axis.

• The point (0,0), where the axes intersect, is called the origin.
• The horizontal line 'y = -2' corresponds to all points where the y-coordinate is -2, regardless of the x-coordinate.
• Points above the x-axis have positive y-values, while those below have negative y-values.

Sketching Graphs

Sketching graphs is a practical way to visually display the solution to an inequality on a coordinate plane. For the inequality \(y < -2\), we represent all solution pairs on the graph by shading the region that meets the inequality's requirement. Here are the pivotal steps to effectively sketch such graphs:

• First, draw the boundary line (in this case \(y = -2\)) as a dashed line to indicate that the points on the line are not included in the solution set.
• Next, determine which side of the line to shade. You can pick a test point that’s not on the line—like (0,0)—to see if it satisfies the inequality. If the test point doesn’t satisfy the inequality, shade the opposite side.
• Remember to label your axes and mark the scale, ideally with evenly spaced increments, to accurately represent the relationship.