Question

Sketch the graph of the inequality. $$5 y \leq-25$$

Short answer

For the given inequality \(5y \leq -25\), the solution after simplification is \(y \leq -5\). This is represented on a number line with a filled dot at -5 and a line extending to the left.

Step-by-step solution

Simplify the Inequality

First, simplify the inequality \(5y \leq -25\). To do that, we want \(y\) by itself on one side of the inequality. We achieve this by dividing both sides of the inequality by 5, which gives \(y \leq -5\).

Draw a Number Line

Create a number line, which will help in visualizing the solution. Draw an arrowhead at both ends of the line to represent that the line extends infinitely in both directions. Ensure to mark the point -5.

Represent the Solution

Since the inequality is \(\leq\) rather than simply \(<\), the solution includes \(y = -5\). We represent this on the number line by a filled dot at -5. As \(y\) is less than or equal to -5, we draw a line extending from the filled dot to the left towards negative infinity.

Key concepts

Solving Inequalities

Inequalities are mathematical expressions involving the symbols <, >, \( \leq \) (less than or equal to), or \( \geq \) (greater than or equal to). The process of solving an inequality is to find all the values of the variable that make the inequality true. Unlike equations, solving inequalities does not yield a single solution, but rather a range of possible solutions.

The first step in solving an inequality is to isolate the variable on one side. This can involve simplifying the expression by adding, subtracting, multiplying, or dividing both sides of the inequality by the same number. When multiplying or dividing by a negative number, remember to reverse the direction of the inequality sign to maintain the correct relationship. For instance, in our exercise, we divided both sides by 5, a positive number, so the direction of the inequality remained the same, and we obtained \( y \leq -5 \).

Number Line Representation

A number line is a valuable tool for visualizing the solutions to inequalities. It consists of a horizontal line with numbers placed at intervals, representing points on the line. To represent the solution to an inequality on a number line:

• First, draw the number line, ensuring it's long enough to include the relevant values.
• Mark the critical value (in this case, -5) with a dot.
• If the inequality is of the type \( \leq \) or \( \geq \) (inclusive), use a filled dot to indicate that the value is part of the solution.
• For the types < or > (strict), use an open dot instead.
• Draw an arrow from the dot towards the direction of the solutions. This could be either left (for less than) or right (for greater than).

In our exercise, after simplifying, we represent \( y \leq -5 \) by placing a filled dot at -5 and drawing a line with an arrow pointing leftward, indicating all the numbers less than -5 are part of the solution.

Inequality Notation

Inequality notation is a system of symbols used to represent the relationship between two values. The symbols used are:

• \( < \): less than
• \( > \): greater than
• \( \leq \): less than or equal to
• \( \geq \): greater than or equal to

Understanding the meaning of these symbols is crucial when solving and graphing inequalities. For instance, the notation \( y \leq -5 \) tells us that \( y \) can take any value less than or equal to -5. This notation is also indicative of which type of dot to use (filled or open) on a number line when graphing the inequality. When using inequality notation to write the solution set, it is common to use interval notation or a number line graph to clearly depict the range of solutions.