Problem 22 - College Algebra with Modeling and Visualization

Question

Solve the inequality symbolically. Express the solution set in set-builder or interval notation. $$ 5-(2-3 x) \leq-5 x $$

Step-by-Step Solution

Answer

Solution: \((-\infty, -\frac{3}{8}]\)

Step 1: Distribute the Negative Sign

First, simplify the left side of the inequality by distributing the negative sign in the term \(-(2-3x)\). This gives:\[ 5 - 2 + 3x \leq -5x \]

Step 2: Combine Like Terms

Next, combine like terms on the left side. Combine constants \(5\) and \(-2\) to get:\[ 3 + 3x \leq -5x \]

Step 3: Move Terms with x to One Side

Add \(5x\) to both sides to get all the \(x\) terms on one side:\[ 3 + 3x + 5x \leq 0 \]

Step 4: Simplify the x Terms

Combine the \(x\) terms on the left side:\[ 3 + 8x \leq 0 \]

Step 5: Isolate x

Subtract \(3\) from both sides to isolate the \(x\) term:\[ 8x \leq -3 \]

Step 6: Solve for x

Divide both sides by \(8\) to solve for \(x\):\[ x \leq -\frac{3}{8} \]

Step 7: Express the Solution

The solution in set-builder notation is:\[ \{ x \mid x \leq -\frac{3}{8} \} \]In interval notation, the solution is:\[ (-\infty, -\frac{3}{8}] \]

Key Concepts

Set-Builder Notation

Set-builder notation is a concise way of describing a set by specifying a property that its members must satisfy. In the context of inequalities, it tells us about the values that a variable can take. For example, if we have a solution set expressed as \( \{ x \mid x \leq -\frac{3}{8} \} \), this is saying "the set of all \( x \) such that \( x \) is less than or equal to \( -\frac{3}{8} \)."
Set-builder notation is useful when you want to be very specific about the elements that belong to a particular set. Here are a few key points:

• The symbol \( \{ \} \) indicates that you are dealing with a set of numbers or elements.
• The vertical bar \( \mid \) means "such that," which introduces the condition after it.
• It allows you to include complex conditions like inequalities, which specify the range of values within the set.

Set-builder notation is particularly helpful in algebra when defining solutions to equations and inequalities in a clear and structured way.

Interval Notation

Interval notation is another method for describing a set of numbers and is often used to express solutions to inequalities. It simplifies the representation of intervals in a more compact form without needing detailed descriptions.
An interval like \( (-\infty, -\frac{3}{8}] \) describes all numbers less than or equal to \( -\frac{3}{8} \). Several key elements make interval notation unique and easy to understand:

• Parentheses \( ( ) \) denote that an endpoint is not included, known as open intervals.
• Brackets \( [ ] \) indicate that an endpoint is included, called closed intervals.
• Infinity symbols \( +\infty \) or \( -\infty \) are used when an interval doesn't have a finite endpoint, and always come with parentheses since infinity itself is not a number.

This method is advantageous in algebra to convey ranges in a simple and standardized manner, making it straightforward to communicate complex solution sets for inequalities.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They stand as the building blocks in solving equations and inequalities.
Let's take a closer look at some components of algebraic expressions:

• **Variables**: Symbols like \( x \) that represent unknown values.
• **Constants**: Specific numbers within an expression like \( 5 \) or \( -2 \).
• **Operators**: The symbols that tell you what to do with the numbers and variables, such as add (+), subtract (-), multiply (\( \cdot \)), and divide (/).

When solving inequalities, algebraic expressions undergo transformations like distribution, combining like terms, and isolating variables to find a solution.
Through these processes, each transformation simplifies the expression, bringing you closer to finding the set of numbers that satisfy the inequality. In the original exercise, you distribute the negative sign and combine like terms to maintain equality while making the equation easier to manipulate. The ultimate goal is to isolate the variable, revealing the solution to the inequality.