Question

Sketch the graph of the inequality. $$8 y>24$$

Short answer

The graph of the inequality \(8y > 24\) is a dashed horizontal line at \(y = 3\), with shading above the line to represent that y-values greater than 3 are included in the solution.

Step-by-step solution

Identifying the Inequality

The inequality to be sketched is \(8y > 24\).

Solve for y

To isolate y, you divide both sides of the inequality by 8. So, \(y > 3\). Since y is greater than 3, that means it could be any value greater than 3.

Sketch the Inequality

On a coordinate plane, draw a horizontal dashed line at \(y = 3\). Because y is greater than 3, shade the region above the line. The dashed line indicates that the values on the line \(y = 3\) are not included in the solution.

Key concepts

Solving Inequalities

Inequalities are mathematical expressions that deal with the relative size or order of two objects or quantities. To solve an inequality means to find all the possible values that satisfy the condition of the inequality. Unlike equations, inequalities do not have a single solution, but rather a range of solutions.

For example, in the inequality 8y > 24, we're looking for all values of y that make the statement true. The first step is to isolate y, which is done by performing the same operation on both sides of the inequality. Dividing both sides by 8, we get y > 3. This means that any number greater than 3 is part of the solution set. It's important to remember that when you divide or multiply both sides of an inequality by a negative number, the direction of the inequality sign changes. However, since we're dividing by a positive number, the inequality sign remains the same.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface with two perpendicular lines called axes. The horizontal axis is typically labeled as the x-axis, and the vertical axis is labeled as the y-axis. The point where these axes intersect is known as the origin, designated as (0,0). Each point on the plane is defined by an ordered pair of numbers (x, y), known as coordinates.

Use of the coordinate plane allows us to visually represent relationships between two variables. Inequalities can be graphed on the coordinate plane to provide a clear picture of the solution set. For example, when graphing the inequality y > 3, we're concerned with the vertical (y-axis) values. The inequality describes a set of points where the y-coordinate is always greater than 3.

Inequality Graphing

When graphing inequalities, the goal is to visually represent all the solutions on the coordinate plane. To graph a simple inequality, such as y > 3, you would start by drawing a line where y = 3. This line serves as a boundary for the inequality. Because the original inequality does not include the line itself (since it's y > 3, not y \(\textgreater=\) 3), we draw a dashed line to indicate that points on the line are not part of the solution.

After the boundary is drawn, the next step is to determine which side of the line to shade. For y > 3, all the solutions lie above the line, so we shade the entire region above the dashed line. This shading represents all points with a y-coordinate greater than 3. It's essential to make sure the shaded area correctly corresponds to the inequality. For example, if the inequality were y < 3, you would shade below the dashed line instead.