Problem 73 - Elementary Algebra

Question

Solve and graph the solution set. In addition, present the solution set in interval notation. $$ 3 x \leq-15 \text { or } 2 x>6 $$

Step-by-Step Solution

Answer

Solution set: \((-\infty, -5] \cup (3, \infty)\).

Step 1: Solve the first inequality

Start by solving the inequality \(3x \leq -15\). Divide both sides of the inequality by 3 to isolate \(x\). The solution becomes \(x \leq -5\).

Step 2: Solve the second inequality

Now, solve the inequality \(2x > 6\). Divide both sides by 2 to isolate \(x\). The solution becomes \(x > 3\).

Step 3: Combine the solutions and express in interval notation

Since we have an 'or' condition, combine the solutions \(x \leq -5\) or \(x > 3\). In interval notation, this is expressed as \((-\infty, -5] \cup (3, \infty)\).

Step 4: Graph the solution set

To graph the solution set on a number line, draw a closed circle at \(-5\) to represent \(x \leq -5\), and shade all numbers to the left. Draw an open circle at 3 and shade to the right to represent \(x > 3\). The shaded parts on both ends of the number line indicate the combined solutions for the inequality.

Key Concepts

Interval Notation

Interval notation is a convenient way of expressing the solution set of an inequality. It provides a clear and concise representation of ranges of numbers. When using interval notation:

• Parentheses \((...,... )\) are used to indicate that an endpoint is not included in the interval. This is known as an open interval.
• Brackets \([..., ... ]\) indicate that an endpoint is included. This is called a closed interval.

For example, in the solution \((-\infty, -5] \cup (3,\infty)\), the square bracket \([-5]\) indicates that -5 is included in the solution, whereas the parentheses around 3 and infinity indicate that they are not included. The union symbol \(\cup\) is used to show that the solution includes all numbers that are either less than or equal to -5, or greater than 3.

Number Line Graph

A number line graph visually represents the solution set of an inequality. It helps to understand the range of values that satisfy the inequality. When graphing the solution set:

• An open circle at a point like 3 indicates that this value is not included in the solution set, as in the inequality \(x > 3\).
• A closed circle indicates inclusion, as in \(x \leq -5\), where a closed circle is placed at -5.

By shading the appropriate section of the number line, we illustrate the complete range of solutions. In our example, one section is shaded to the left of -5, including -5, and another section to the right of 3, excluding 3. These shaded areas represent all the solutions for the compound inequality.

Compound Inequality

A compound inequality involves two distinct inequalities joined by "and" or "or." It defines a condition where at least one or both parts of the inequality must be true.

• If connected with "and," both conditions must be satisfied simultaneously.
• If connected with "or," satisfying either inequality makes the entire statement true. This is the case in the example \(3x \leq -15 \text{ or } 2x > 6\).

Here, you solve each inequality separately and then look at the union of their solution sets. Our solution \((x \leq -5) \text{ or } (x > 3)\) captures numbers satisfying at least one of the inequalities, representing all possible solutions.

Solution Set Representation

Representing a solution set effectively communicates the solution to inequalities. It can be expressed in multiple formats such as algebraic expressions, interval notation, or graphically on a number line. For our example:

• Algebraically, it's shown as \(x \leq -5\) or \(x > 3\).
• Interval notation presents it as \((-\infty, -5] \cup (3,\infty)\).
• The number line graph visualizes it with shaded areas and open/closed circles.

Each representation provides a different perspective but conveys the same information: all values less than or equal to -5, or greater than 3 are solutions to the compound inequality.