Problem 15 - Algebra and Trigonometry

Question

\(9-32\) me solve the linear inequality. Express the solution using interval notation and graph the solution set. $$ 2 x+1<0 $$

Step-by-Step Solution

Answer

The solution is \((-\infty, -\frac{1}{2})\).

Step 1: Isolate the Variable Term

To solve the inequality, we first need to isolate the term with the variable. We start with the inequality \(2x + 1 < 0\). We subtract 1 from both sides to get:\[2x < -1\]

Step 2: Solve for the Variable

Next, we need to solve for \(x\). Since \(x\) is being multiplied by 2, we divide both sides by 2:\[x < -\frac{1}{2}\]

Step 3: Write the Solution in Interval Notation

In interval notation, \(x < -\frac{1}{2}\) is written as:\((-\infty, -\frac{1}{2})\)

Step 4: Graph the Solution Set

To graph the solution, draw a number line and shade to the left of -\(\frac{1}{2}\). Place an open circle at -\(\frac{1}{2}\) to indicate that this value is not included in the solution.

Key Concepts

Interval Notation

Interval notation is a mathematical shorthand used for representing a range of numbers, which is particularly useful when dealing with inequalities. In our exercise, the inequality we solved was \(x < -\frac{1}{2}\). In this case, we express the solution in interval notation as \(( -\infty, -\frac{1}{2} )\). This tells us that all numbers less than \(-\frac{1}{2}\) are part of the solution set. The parentheses \(( )\) indicate that \(-\frac{1}{2}\) itself is not part of the set, as opposed to square brackets \([ ]\) which would mean otherwise. Additionally, because the range extends indefinitely in the negative direction, we use \(-\infty\) to signify that there is no lower bound.

Solution Set Graph

The solution set graph visually represents the solutions of an inequality on a number line. Once we determined that the solution to \(2x + 1 < 0\) is \(x < -\frac{1}{2}\), we translated this solution into a graph.
To do so, we place an open circle at \(-\frac{1}{2}\) on the number line. The open circle signals that the endpoint \(-\frac{1}{2}\) is not part of the solution. We then shade the portion of the line extending to the left of \(-\frac{1}{2}\).
This shading represents all numbers less than \(-\frac{1}{2}\), which are included in the solution set.

• Use open circles for strict inequalities like \(<\) and \(>\)
• Shade in the direction that represents the solution set

Graphing helps visualize which numbers satisfy the inequality, making interpretation easier.

Solving Inequalities

Solving inequalities involves finding all possible values of the variable that make the inequality true. The process is similar to solving equations, but with a key exception: direction of the inequality symbol matters, especially when multiplying or dividing by a negative number.
In our original problem, we began by isolating the variable term by subtracting 1 from both sides, simplifying the inequality to \(2x < -1\). Then, we divided both sides by 2 to solve for \(x\). This gave us \(x < -\frac{1}{2}\).
Some key points to remember when solving inequalities:

• Perform the same operation on both sides to maintain the inequality
• If you multiply or divide by a negative number, flip the inequality sign

By keeping these rules in mind, you can solve inequalities systematically and accurately.