Problem 22 - Precalculus Mathematics for Calculus

Question

Solve the linear inequality. Express the solution using interval notation and graph the solution set. $$\frac{2}{5} x+1<\frac{1}{5}-2 x$$

Step-by-Step Solution

Answer

x < -1/3; Interval: (-∞, -1/3)

Step 1: Eliminate Fractions

Multiply every term in the inequality \( \frac{2}{5}x + 1 < \frac{1}{5} - 2x \) by 5 to eliminate the fractions. This results in the inequality \( 2x + 5 < 1 - 10x \).

Step 2: Collect Variable Terms on One Side

Add \( 10x \) to both sides of the inequality to collect all the \( x \) terms on one side. The inequality becomes \( 12x + 5 < 1 \).

Step 3: Isolate the Variable Term

Subtract 5 from both sides to isolate the \( x \) term. This results in \( 12x < -4 \).

Step 4: Solve for x

Divide both sides of the inequality by 12 to solve for \( x \), resulting in \( x < -\frac{1}{3} \).

Step 5: Express Solution in Interval Notation

The solution \( x < -\frac{1}{3} \) can be expressed in interval notation as \( (-\infty, -\frac{1}{3}) \).

Step 6: Graph the Solution Set

On a number line, draw an open circle at \(-\frac{1}{3}\) and shade the line to the left to represent all numbers less than \(-\frac{1}{3}\).

Key Concepts

Interval Notation

Interval notation provides a concise way to represent a set of numbers between two endpoints. For linear inequalities, it's handy for expressing solution sets without listing every element explicitly.

With interval notation, we use brackets or parentheses to describe inclusion or exclusion:

• "(" and ")" are used when the endpoints are not included, called open intervals. For example, \(a, b\) includes all numbers greater than \(a\) and less than \(b\).
• "[" and "]" are used when endpoints are included, forming closed intervals. For instance, \[a, b\] includes all numbers from \(a\) to \(b\) and includes the endpoints.

In our example, the solution to the inequality is \(x < -\frac{1}{3}\). This indicates all values of \(x\) that are less than \(-\frac{1}{3}\). Because \(-\frac{1}{3}\) is not included in the solution, we use a parenthesis: \((-\infty, -\frac{1}{3})\).

The symbol \(-\infty\) signifies that the interval continues indefinitely in the negative direction, and it always uses a parenthesis because infinity is not a concrete number we can include.

Graphing Inequalities

Graphing inequalities allows us to visually represent solutions on a number line. This method helps illustrate which parts of the line are included in the solution set.

To graph the inequality \(x < -\frac{1}{3}\), follow these steps:

• First, mark the point corresponding to \(-\frac{1}{3}\) on the number line.
• Since \

Solving Inequalities

Solving linear inequalities is very similar to solving equations, but there's an important distinction to remember: multiplying or dividing by a negative number flips the inequality sign.

Here's how to solve an example inequality:1. **Eliminate Fractions:** To simplify, multiply every term of the inequality \(\frac{2}{5}x + 1 < \frac{1}{5} - 2x\) by 5. This clears fractions, making calculations easier. The new expression becomes \(2x + 5 < 1 - 10x\).2. **Get All Variables on One Side:** Transfer all \(x\) terms to one side by adding \(10x\). This simplifies the inequality to \(12x + 5 < 1\).3. **Isolate the \(x\) Term:** To get \(x\) by itself, subtract 5 from both sides, resulting in \(12x < -4\).4. **Solve for \(x\):** Divide every term by 12 to find that \(x < -\frac{1}{3}\).

Each step helps simplify the inequality, bringing \(x\) to one side and numbers to the other. Remember that whenever we multiply or divide by a negative, we must reverse the inequality symbol. This rule is crucial for maintaining accuracy when solving inequalities.