Problem 75 - Elementary Algebra

Question

Solve and graph the solution set. In addition, present the solution set in interval notation. $$ -2 x+1<-1 \quad \text { or }-2 x+1>1 $$

Step-by-Step Solution

Answer

The solution set is \((-\infty, 0) \cup (1, \infty)\).

Step 1: Solve the first inequality

First, solve the inequality \(-2x + 1 < -1\). Start by subtracting 1 from both sides to isolate the term with \(x\):\[-2x + 1 - 1 < -1 - 1\]This simplifies to:\[-2x < -2\]Next, divide both sides by \(-2\). Remember to reverse the inequality sign since we are dividing by a negative number:\[x > 1\]

Step 2: Solve the second inequality

Now solve the inequality \(-2x + 1 > 1\). Similarly, subtract 1 from both sides:\[-2x + 1 - 1 > 1 - 1\]This simplifies to:\[-2x > 0\]Divide both sides by \(-2\) and reverse the inequality sign:\[x < 0\]

Step 3: Interpret the solution set

The solution to the inequality \(-2x + 1 < -1\) is \(x > 1\), and the solution to \(-2x + 1 > 1\) is \(x < 0\). Since the problem states 'or', the solution set includes all \(x\) that satisfy either inequality. This means \(x\) can be greater than 1 or less than 0.

Step 4: Express the solution in interval notation

In interval notation, the solution \(x > 1\) is represented as \((1, \infty)\), and \(x < 0\) is represented as \((-\infty, 0)\). Combining these using the 'or' condition gives:\((-\infty, 0) \cup (1, \infty)\).

Step 5: Graph the solution set

On a number line, represent the interval \((-\infty, 0)\) as an arrow extending to the left from 0, not including 0, and \((1, \infty)\) as an arrow extending to the right from 1, not including 1. Use open circles at 0 and 1 to indicate that these points are not included in the solution set.

Key Concepts

Solution Set

Understanding the solution set is key when working with inequalities. It refers to all the possible values of the variable, usually represented as "x", that satisfy the given inequality. In this problem, the inequalities are:

• For \(-2x + 1 < -1\), solving this gives us \(x > 1\).
• For \(-2x + 1 > 1\), the solution is \(x < 0\).

This means the solution set includes all values of x that are either greater than 1 or less than 0. It's important to recognize that the "or" in this context means any value that satisfies at least one of the inequalities is part of the solution set.
In problems where multiple conditions are combined, always check if the connector is "and" or "or," as they lead to different solution sets.

Interval Notation

Interval notation is a way of describing the solution set of an inequality in a compact form. It uses parentheses and brackets to show which numbers are included or excluded. Here’s how to break it down:

• A parenthesis, \(\text{(}\text{ or }\text{)}\), means the number is not included, which is known as "open".
• A bracket, \[\text{[}\text{ or }\text{]}\], means the number is included, which is known as "closed".

For the inequalities \(x > 1\) and \(x < 0\), the interval notation becomes:

• For \(x > 1\), write as \( (1, \infty)\), indicating all numbers greater than 1.
• For \(x < 0\), write as \( (-\infty, 0)\), indicating all numbers less than 0.

When combined with "or," the solution is expressed as \( (-\infty, 0) \cup (1, \infty)\). The union symbol \(\cup\) indicates that values from both intervals are part of the solution.

Graphing Inequalities

Graphing inequalities visually represents the solution set on a number line or coordinate plane. To graph the given problem:
Start with a number line.

• For \(x < 0\), draw an arrow extending to the left from 0, using an open circle at 0 to show that it’s not included.
• For \(x > 1\), draw an arrow extending to the right from 1, with an open circle at 1 to indicate exclusion.

This graph will show two separate sections, one on the left of 0 and the other on the right of 1, illustrating the solution set \( (-\infty, 0) \cup (1, \infty)\). Graphs can be a powerful tool to visually verify the solution and easily understand how the intervals are spread.