Problem 62 - College Algebra with Modeling and Visualization

Question

Solve the inequality. Write the solution in interval notation. $$|5 x-7|>2$$

Step-by-Step Solution

Answer

The solution is \((-\infty, 1) \cup \left(\frac{9}{5}, \infty\right)\).

Step 1: Understand the Absolute Value Inequality

The inequality given is \(|5x-7| > 2\). This type of inequality can be broken into two separate inequalities when dealing with the absolute value. Hence, it becomes two scenarios: \(5x - 7 > 2\) or \(5x - 7 < -2\). We will solve each inequality separately.

Step 2: Solve the Positive Inequality

Let's solve the first part \(5x - 7 > 2\). Add 7 to both sides to isolate the term with \(x\):\[5x > 2 + 7\]\[5x > 9\]Then, divide both sides by 5:\[x > \frac{9}{5}\]

Step 3: Solve the Negative Inequality

Now solve the second part \(5x - 7 < -2\). Add 7 to both sides:\[5x < -2 + 7\]\[5x < 5\]Now, divide both sides by 5:\[x < 1\]

Step 4: Combine the Solutions

The solution to \(|5x-7| > 2\) comes from combining the results of our two inequalities. From the first inequality, \(x > \frac{9}{5}\). From the second inequality, \(x < 1\). Since both conditions are needed, the solution is a combination of these intervals: \(x < 1\) or \(x > \frac{9}{5}\).

Step 5: Express in Interval Notation

The final solution in interval notation is the union of two intervals: \((-\infty, 1) \cup \left(\frac{9}{5}, \infty\right)\). This signifies all numbers less than 1, along with all numbers greater than \(\frac{9}{5}\).

Key Concepts

Interval Notation

Interval notation is a streamlined way to represent the solution set of an inequality. Instead of listing individual numbers, interval notation uses parentheses and brackets to describe all the numbers that satisfy the inequality.

• Parentheses \( ( \, \text{and} \, ) \) are used when the endpoints of the set are not included (open intervals).
• Brackets \( [ \, \text{and} \, ] \) signify that the endpoints are included in the set (closed intervals).

For example, in our problem, the solution \(x<1\) or \(x>\frac{9}{5}\) translates to the interval notation as \((−\infnty, 1) \, \cup \, \left(\frac{9}{5}, \infty\right)\). This indicates a combination of numbers starting from negative infinity up to 1, excluding 1, and from \(\frac{9}{5}\) onwards, also excluding \(\frac{9}{5}\). The union symbol \(\cup\) is used to join these two intervals.

Inequality Solving

Solving inequalities often involves multiple steps, including isolating the variable and splitting absolute value inequalities into two cases. Here are the basic steps:

1. Rewrite \( |expression| > c \) as two separate inequalities: \( expression > c \) and \( expression < -c \).
2. For each inequality, isolate the variable. This will usually involve adding, subtracting, multiplying, or dividing both sides by the same number.
3. Remember to flip the inequality sign anytime you multiply or divide by a negative number.

In our exercise, two inequalities arise from \( |5x-7|>2 \): \( 5x - 7 > 2 \) and \( 5x - 7 < -2 \). Solving these gives \( x > \frac{9}{5} \) and \( x < 1 \,\), respectively.

Algebraic Expressions

An algebraic expression consists of variables, numbers, and operations. It forms the core of equations and inequalities like those in our problem.

• An expression like \(5x - 7\) involves a variable \(x\) multiplied by 5, with 7 subtracted from the product.
• The expression must be manipulated carefully to maintain equality or inequality while solving mathematical problems.
• Operations include addition, subtraction, multiplication, and division, allowing one to move terms across the equal sign or inequality while preserving the relationships between terms.

For instance, solving \(5x - 7 > 2\) involves adding 7 to both sides to manage the constant terms, then dividing by 5 to solve for \(x\). Understanding how to manipulate such expressions is essential in solving inequalities and writing them in a more interpretable form like interval notation.