Problem 11 - College Algebra with Modeling and Visualization

Question

Solve the inequality symbolically. Express the solution set in set-builder or interval notation. $$ -2(x-10)+1>0 $$

Step-by-Step Solution

Answer

The solution is \\(x < \frac{21}{2}\), or in interval notation, \\((-\infty, \\\frac{21}{2})\\).\\).

Step 1: Expand the Expression

Start by expanding the expression on the left-hand side of the inequality. Distribute the \(-2\) across the \(x-10\) term: \\(-2(x-10) = -2x + 20\). This changes the inequality to \(-2x + 20 + 1 > 0\).

Step 2: Simplify the Expression

Combine like terms on the left-hand side of the inequality. The expression \\(-2x + 20 + 1\) simplifies to \(-2x + 21\). Now the inequality is \(-2x + 21 > 0\).

Step 3: Move Constants to the Other Side

Subtract 21 from both sides of the inequality to isolate the term with \(x\): \\(-2x > -21\).

Step 4: Isolate the Variable

Divide both sides of the inequality by \(-2\) to solve for \(x\). Remember, dividing by a negative number reverses the inequality sign: \(\x < \frac{21}{2}\).

Step 5: Express Solution in Interval Notation

The inequality \(x < \frac{21}{2}\) is expressed in interval notation as \((-\infty, \frac{21}{2})\).

Key Concepts

Set-builder notation

Set-builder notation is a powerful way to describe a set by specifying a property that its members must satisfy. It's a mathematical shorthand that provides a concise way of expressing sets, especially in mathematics and higher-level algebra. In set-builder notation, the solution to an inequality is written in a form that explicitly describes the conditions for a variable to be a part of the set.

For the inequality problem we solved, we ended with the inequality: \( x < \frac{21}{2} \). In set-builder notation, this is expressed as:

• \( \{ x \mid x < \frac{21}{2} \} \)

This notation reads as "the set of all \( x \) such that \( x \) is less than \( \frac{21}{2} \)."

The vertical bar "\(|\)" is used to mean "such that," making it clear what properties the elements of the set must satisfy. This clarity is one reason set-builder notation is favored when describing complex sets or solutions.

Interval notation

Interval notation is an efficient method of denoting a range of values on the number line. It uses simple symbols to express a set of numbers that satisfy an inequality. In our solved exercise, we end with the inequality \( x < \frac{21}{2} \).

In interval notation, this is written as:

• \(( -\infty, \frac{21}{2} )\)

The rounded parenthesis \(()\) indicate that \( -\infty \) and \( \frac{21}{2} \) are not included in the set. This contrasts with square brackets \([]\), which would mean inclusion.

By using interval notation, mathematicians and students alike can quickly convey the solution to inequalities in a compact form. This method simplifies reading and interpreting mathematical information, especially when dealing with continuous sets of numbers.

Variable isolation

Variable isolation is a crucial step in solving equations and inequalities. It involves manipulating an equation or inequality so that the variable is alone on one side of the equation. This gives a clear expression for the solution. In our exercise, we started with the inequality: \[-2x + 21 > 0\].
The process of variable isolation includes several steps:

• First, constants are moved to the other side of the inequality by performing inverse operations. For example, subtracting 21 from both sides results in \(-2x > -21\).
• Next, to isolate \( x \), we divided by \(-2\), remembering to reverse the inequality to \( x < \frac{21}{2} \).

When dividing or multiplying both sides of an inequality by a negative number, it's essential to reverse the inequality sign. This rule helps maintain the inequality's direction and accuracy.

Successfully isolating variables is a foundational skill in algebra that simplifies complex equations and inequalities, allowing students to find solutions more intuitively and accurately.