Problem 20 - Algebra and Trigonometry

Question

\(9-32\) me 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

The solution is \((-\infty, -\frac{1}{3})\).

Step 1: Isolate the variable on one side

We start with the inequality: \( \frac{2}{5}x + 1 < \frac{1}{5} - 2x \).To bring terms involving \(x\) to one side, add \(2x\) to both sides:\( \frac{2}{5}x + 1 + 2x < \frac{1}{5} \).Simplify the left side by converting \(2x\) into the same fraction form:\( \frac{2}{5}x + \frac{10}{5}x + 1 < \frac{1}{5} \).Combine like terms:\( \frac{12}{5}x + 1 < \frac{1}{5} \).

Step 2: Eliminate the constant term

Subtract 1 from both sides to eliminate the constant on the left:\( \frac{12}{5}x + 1 - 1 < \frac{1}{5} - 1 \).This simplifies to:\( \frac{12}{5}x < \frac{1}{5} - \frac{5}{5} \).Which becomes:\( \frac{12}{5}x < -\frac{4}{5} \).

Step 3: Solve for the variable

To solve for \(x\), multiply both sides by the reciprocal of \(\frac{12}{5}\), which is \(\frac{5}{12}\):\(x < -\frac{4}{5} \times \frac{5}{12} \).Simplify this by multiplying the fractions:\(x < -\frac{4 imes 5}{5 imes 12} = -\frac{20}{60} \).Finally, simplify the fraction:\( x < -\frac{1}{3} \).

Step 4: Express the solution in interval notation

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

Step 5: Graph the solution set

To graph the solution \((-\infty, -\frac{1}{3})\), draw a number line.Mark \(-\frac{1}{3}\) with an open circle to indicate that \(-\frac{1}{3}\) is not included in the solution set.Shade the line to the left of \(-\frac{1}{3}\), extending indefinitely, to represent all numbers less than \(-\frac{1}{3}\).

Key Concepts

Interval Notation

Interval notation is a convenient way to describe a range of numbers without using inequalities. It uses brackets and parentheses to specify the endpoints of an interval. For instance, the inequality solution \( x < -\frac{1}{3} \) is expressed in interval notation as \((-\infty, -\frac{1}{3})\). Here’s how interval notation works:

• Parentheses \(( )\) indicate that the endpoint is not included, also known as an open interval. For example, \((-\infty, -\frac{1}{3})\) shows that \(-\frac{1}{3}\) is not part of the solution.
• Brackets \([ ]\) mean the endpoint is included, indicating a closed interval.
• The symbol \(-\infty\) always pairs with a parenthesis since infinity can never be reached or included.

Interval notation offers a concise representation of solutions, making it easier to read and interpret inequalities.
Understanding this notation is a valuable skill for solving mathematical problems efficiently.

Number Line

A number line is a visual representation of numbers laid out on a straight line, which helps to illustrate numerical relationships, such as inequalities. In the context of linear inequalities, it is a handy tool to demonstrate solution sets. Let's consider the inequality \( x < -\frac{1}{3} \).

To graph this inequality on a number line:

• Certain terms, such as the endpoint \(-\frac{1}{3}\), are marked. In this case, use an open circle to signify that \(-\frac{1}{3}\) is not included in the solution.
• Shade the portion of the line extending to the left of \(-\frac{1}{3}\). This shaded region signifies all numbers less than \(-\frac{1}{3}\).

The number line serves as a visual verification of the solution, making abstract concepts more tangible.
It helps build an intuitive understanding of the range of values satisfying the inequality.

Fraction Simplification

Fraction simplification is essential when solving linear inequalities, as it reduces complexity and clarifies the expression. In the original problem, simplifying fractions was crucial to isolate and solve for \(x\). Here’s a step-by-step look at how to simplify the fraction \(-\frac{20}{60}\) to \(-\frac{1}{3}\).

Simplifying fractions involves:

• Identifying the greatest common divisor (GCD) of both the numerator and the denominator. For \(20\) and \(60\), the GCD is \(20\).
• Dividing both the numerator and the denominator by this GCD. So, \(\frac{20}{60}\) simplifies to \(\frac{1}{3}\) because \(20 \div 20 = 1\) and \(60 \div 20 = 3\).

Consistently applying fraction simplification results in more accessible and concise mathematical expressions.
It is a critical step not only for solving equations but also for ensuring that solutions are presentable and easier to interpret.