Problem 78 - College Algebra with Modeling and Visualization

Question

Solve each inequality numerically. Write the solution set in set-builder or interval notation, and approximate endpoints to the nearest tenth when appropriate. $$ 1.5(x-0.7)+1.5 x<1 $$

Step-by-Step Solution

Answer

The solution set is \((-\infty, 0.7)\).

Step 1: Distribute the 1.5

Multiply 1.5 by each term inside the parentheses and add 1.5x to both sides: \[ 1.5 \times x - 1.5 \times 0.7 + 1.5x < 1 \]Simplifies to:\[ 1.5x - 1.05 + 1.5x < 1 \]

Step 2: Combine like terms

Combine the terms with \(x\):\[ 1.5x + 1.5x - 1.05 < 1 \]This results in:\[ 3x - 1.05 < 1 \]

Step 3: Isolate the variable term

Add 1.05 to both sides to get the term with \(x\) by itself:\[ 3x - 1.05 + 1.05 < 1 + 1.05 \]Which simplifies to:\[ 3x < 2.05 \]

Step 4: Solve for x

Divide each side by 3 to solve for \(x\):\[ x < \frac{2.05}{3} \]Calculate the division:\[ x < 0.6833 \]

Step 5: Write the solution in interval notation

Since \(x\) is less than approximately 0.7 (rounded to the nearest tenth), the numbers satisfying the inequality are all numbers less than 0.7. In interval notation, this is:\[ (-\infty, 0.7) \].

Key Concepts

Set-builder notation

Set-builder notation is a way to express a set by describing the properties that its members must satisfy. It is particularly useful in mathematics when defining sets with conditions or constraints. For example, we might express a set using set-builder notation as \( \{ x \mid x < 0.7 \} \). This reads as "the set of all \( x \) such that \( x \) is less than 0.7."

• Starts with a curly brace \( \{ \)
• Includes a variable, often \( x \), followed by a vertical bar \( \mid \) or a colon \( : \)
• States the condition or inequality that members of the set must satisfy

This method is a concise way to show the solution set of an inequality, and it succinctly communicates the same idea that interval notation might represent using different symbols.

Interval notation

Interval notation is a shorthand used to describe a range of numbers that satisfy a certain condition or inequality. It uses brackets and parentheses to express the start and end of an interval:

• Round brackets, \((\) and \()\), denote that the endpoint is not included (an open interval).
• Square brackets, \([\) and \(]\), indicate that the endpoint is included (a closed interval).

For our example, where \( x < 0.7 \), the interval notation is \((-\infty, 0.7)\). This shows all numbers less than 0.7, starting from negative infinity, without including 0.7 itself.
Interval notation is particularly useful in quickly conveying ranges without extra words, making it efficient in mathematical writing and solutions.

Numerical approximation

Numerical approximation involves estimating a number to a certain degree of accuracy or precision. This often occurs in solving equations or inequalities where exact arithmetic isn't feasible or necessary. When we approximate, especially in the context of solving inequalities, we aim to get close to the actual value but limit our digits to what's practically useful.
For example, when solving the inequality \( 3x < 2.05 \), dividing to compute \( x < \frac{2.05}{3} \) gives us \( x < 0.6833 \).
Rounding this to the nearest tenth, we use 0.7 for practical simplicity and when endpoint precision isn't critical. Numerical approximation aids in simplifying complex calculations, especially when providing a final solution that's easy to interpret.

Solving inequalities

Solving inequalities involves finding all values of the variable that satisfy the inequality condition. Inequalities are similar to equations but with signs like \(<\), \(>\), \(\leq\), or \(\geq\) instead of an equal sign.
The solving process often includes:

• Simplifying both sides of the inequality by distributing and combining like terms.
• Isolating the variable by adding or subtracting terms and then dividing or multiplying as necessary.

In our example, beginning with \( 1.5(x-0.7) + 1.5x < 1 \), we executed several steps to isolate \( x \):

• Distributed the 1.5 across the terms.
• Combined like terms to simplify.
• Added values to both sides to isolate the term with \( x \).
• Finally divided by 3 to solve for \( x \).

The outcome, \( x < 0.7 \), reveals all values making the inequality true. Inequalities often lead to ranges of solutions, which we express efficiently using set-builder or interval notation.