Question

Sketch the graph of the inequality. $$6 y \leq 24$$

Short answer

The graph of the inequality \(6y \leq 24\) is a horizontal line at \(y = 4\) and the area below this line is shaded to represent the solution to the inequality.

Step-by-step solution

Solve for y

To solve for \(y\), divide both sides of the inequality \(6y \leq 24\) by 6. This gives \(y \leq 4\).

Draw a Horizontal Line

Next, draw a horizontal line on the graph at \(y = 4\). This line represents the inequality \(y \leq 4\).

Shade the Region

As y is less than or equal to 4, shade the region below or the region that includes the line y=4. This shaded region represents all possible solutions to the inequality.

Key concepts

Solving Inequalities

Understanding how to solve inequalities is foundational in algebra. Unlike equations, inequalities indicate that one side is not exactly equal to, but rather less than (<), less than or equal to (\textless=), greater than (>), or greater than or equal to (\textgreater=) the other side. To solve an inequality, such as \(6y \leq 24\), we perform similar steps as we would with an equation, keeping a key rule in mind: when we multiply or divide by a negative number, we must flip the inequality sign.

For the inequality \(6y \leq 24\), we aim to isolate the variable \(y\). We start by dividing both sides by 6, resulting in \(y \leq 4\), which tells us \(y\) can be any number up to and including 4. It is crucial to divide both sides of the inequality by the same positive number to maintain the direction of the inequality sign.

Inequality Graphing

Graphing inequalities involves representing the set of all possible solutions on a number line or coordinate plane. For the inequality \(y \leq 4\), we start by identifying the boundary of our solution set, which is \(y = 4\). In a two-dimensional graph, this boundary is represented by a horizontal line, because all points along this line have the same \(y\)-value.

It is essential to use a solid line to denote \(y \leq 4\) since the inequality includes the value of 4 itself. If the inequality were strict (\(<\) or \(>\)), we would use a dashed line instead to show that the line is not part of the solution set. After drawing the boundary line, the next step is shading the solution region.

Shading Solution Regions

The shading of solution regions in graphing inequalities represents all the possible values that satisfy the inequality. For the inequality \(y \leq 4\), we shade the region below the horizontal line, because we are looking for all \(y\)-values that are 'less than or equal to' 4.

It's important to shade the correct side of the line to accurately represent the inequality. If you're ever unsure which side to shade, pick a test point not on the line, like (0,0), and see if it satisfies the inequality. If it does, shade that side; if not, shade the other side. As \(0 \leq 4\) is true, the side containing \(y = 0\) is shaded. This visual representation is pivotal in helping us see the range of possible solutions to an inequality.

Horizontal Line in Graph

A horizontal line on a graph has a constant \(y\)-value across all \(x\)-values, which is why it is instrumental in graphing the inequality \(y = 4\) for \(y \leq 4\). The idea is that every point on this line shares the same \(y\)-value, in this case, 4. In a graphical sense, this line is crucial because it represents the exact boundary where the inequality’s conditions are met—at every point along this line, the \(y\)-value is 4, which is the maximum value of \(y\) allowed by the inequality.

Thus, for every point to the left or right on this line, \(y\) remains at 4, demonstrating that the \(x\)-value has no effect on the inequality. The horizontal line creates a visual demarcation in the graph that aids in distinguishing the solution region which, in this case, is the area extending infinitely below the line.