Problem 53 - College Algebra with Modeling and Visualization

Question

Use the \(x\) -intercept method to solve the inequality. Write the solution set in set-builder or interval notation. Then solve the inequality symbolically. $$ 2-x<3 x-2 $$

Step-by-Step Solution

Answer

The solution is \(x > 1\), or in interval notation: \((1, \infty)\).

Step 1: Rearrange the Inequality

Start by moving all the terms to one side of the inequality. Subtract \(3x\) from both sides: \[ 2 - x - 3x < -2 \] Simplify by combining like terms: \[ 2 - 4x < -2 \]

Step 2: Isolate the Variable

Continue by isolating \(x\) on one side. Subtract 2 from both sides: \[ -4x < -4 \] Now, divide each side by \(-4\), remembering to flip the inequality sign: \[ x > 1 \]

Step 3: Write the Solution

Express the solution \(x > 1\) in both interval notation and set-builder notation.- Interval notation: \((1, \infty)\)- Set-builder notation: \( \{ x \mid x > 1 \} \)

Step 4: Verify the Solution

Check if this interval satisfies the original inequality. Choose a value of \(x\) within the solution interval, such as \(x = 2\). Substitute \(x = 2\) into the original inequality: \[ 2 - 2 < 3 \times 2 - 2 \] Simplify: \[ 0 < 4 \] The inequality holds true, verifying that the solution \(x > 1\) is correct.

Key Concepts

Understanding the x-intercept method

The x-intercept method is a practical approach to solving inequalities by finding where the expression equals zero. Think of it as finding the "break-even" point where the inequality changes from less than or greater than to the other side. To use this method, we first need to rearrange the inequality as follows:

• Move all terms to one side, making one side equal to zero.
• For example, in the inequality given: \(2-x<3x-2\), subtracting \(3x\) from both sides gives us: \(2-x-3x < -2\).
• Combine like terms resulting in \(2-4x < -2\).
• Once in standard form (\(f(x) = 0\)), find the x-intercept by setting the expression equal to zero.

This method gives us key insights into where the inequality changes direction, helping us set the stage for solving it symbolically or using other notations.

Exploring set-builder notation

Set-builder notation is a formal way of expressing a collection of numbers or objects that satisfy a specific condition. This mathematical language is particularly useful when describing solutions to inequalities. For the inequality \(x > 1\), we can express the solution in set-builder notation as \(\{ x \mid x > 1 \}\). This notation consists of:

• A variable, here \(x\), representing elements of the set.
• A vertical bar or colon, \(|\), meaning "such that".
• A condition, \(x > 1\), specifies the criteria members of the set must satisfy.

Set-builder notation allows us to communicate precisely which values belong to our solution set. It is particularly useful for algebraic descriptions, offering an alternative to graphical representations or other forms like interval notation.

How interval notation simplifies solutions

Interval notation is a concise way to describe ranges of values that satisfy an inequality. It's often used in conjunction with set-builder notation to clearly convey solution sets. In interval notation, the solution \(x > 1\) is represented as \((1, \infty)\). Let's break that down:

• The parenthesis \((\) indicates that 1 is not included in the solution, meaning the values are "greater than" but not "equal to" 1.
• \(\infty)\) suggests that the solution extends indefinitely in the positive direction, meaning all numbers greater than 1 satisfy the inequality.

Interval notation is highly efficient as it quickly conveys the extent of valid values without verbose explanation. It's broadly used in calculus and higher mathematics, making it a key component in expressing solutions succinctly.

Symbolic solution methods

Solving an inequality symbolically involves manipulating the inequality using algebraic principles until the variable is isolated. This transforms the inequality into a simple expression or statement of the solution. For our inequality \(2-x < 3x-2\), the symbolic solution follows these steps:

• Move all \(x\) terms to one side, getting \(2 - 4x < -2\).
• Isolate \(x\) by subtracting 2, resulting in \(-4x < -4\).
• Divide each term by \(-4\) while flipping the inequality sign (a key rule when dividing by a negative number), resulting in \(x > 1\).

Symbolic methods provide a straightforward way to obtain the exact solution, often forming the foundation for further analysis with other notations or graphing.