Problem 45 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the inequality and sketch the solution on the real number line. Use a graphing utility to verify your solutions graphically. $$10|1-x|<5$$

Step-by-Step Solution

Answer

The solution to the inequality \(10|1-x|<5\) is 0.5 <= x <= 1.5.

Step 1: Simplify the inequality

Firstly, we want to simplify the inequality by isolating the absolute value term. To do this, we divide both sides by 10 to result in \(|1-x|<0.5\).

Step 2: Write positive and negative equations

Remembering the definition of absolute value, we set up two equations to solve, one for the positive equivalence and the other for the negative, resulting in two inequalities 1 - x = 0.5 (where we assume x < 1) and 1 - x = -0.5 (where we assume x > 1).

Step 3: Solve both inequalities

Solving the inequalities (1-x = 0.5 and 1 - x = -0.5) respectively gives two solutions x = 0.5, and x = 1.5.

Step 4: Check and confirm the solutions

We then confirm the solutions by substituting these values for x back into the original inequality. For values 0.5<= x <=1.5, the inequality \(|1-x|<0.5\) holds true.

Step 5: Graphically confirm the solutions

To graphically confirm the solution, plot the line y = 1 - x (the absolute value has been omitted for simplicity) and the line y = 0.5 in the same graph. The region between these two lines represents the solutions. In a graphing utility, the x-values where y = 0.5 is below y = 1 - x coincide with the solution interval identified previously, confirming the solution.

Step 6: Final Solution

Therefore, the solution for the inequality \(10|1-x|<5\) is 0.5 <= x <= 1.5.

Key Concepts

Absolute Value Inequality

An absolute value inequality involves an expression within an absolute value bracket, indicating how far a number is from zero on the number line. In this case, we are working with the inequality \( |1-x| < 0.5 \). The absolute value notation \(|...|\) means you consider both positive and negative scenarios. Hence, solving \( |1-x| < 0.5 \) requires solving two different inequalities:

• \( 1-x < 0.5 \)
• \( 1-x > -0.5 \)

This enables us to consider both sides of the inequality. Once broken down, each inequality can be solved for the variable \(x\) to find the set of possible solutions that satisfy the original absolute value inequality.

Real Number Line

The real number line is a straight line that represents all possible real numbers. In mathematics, this concept is used to visually represent solutions to inequalities. For the inequality \( 10|1-x|<5 \), the solution represents all values of \(x\) that satisfy the condition when plotted on this line.
Here's how you do it:

• You solve the inequality to get the range of solutions (\(0.5 \leq x \leq 1.5\)).
• Plot these values on the number line.

On the number line, you draw a shaded region or a bold line segment between 0.5 and 1.5 to indicate all possible solutions. It gives you a visual confirmation of the values where the inequality holds true.

Graphing Utility

A graphing utility is a tool, such as a graphing calculator or computer software, which helps visualize mathematical equations and their solutions. For the inequality \(10|1-x|<5\), a graphing utility can plot the function \(y = |1-x|\) and the line \(y = 0.5\) (after simplifying the inequality to \(|1-x|<0.5\)).
By graphing these, you seek the intervals where the graph of \(y = |1-x|\) is below the horizontal line \(y = 0.5\). This visual check confirms the solution range we found algebraically (\(0.5 \leq x \leq 1.5\)). Graphing utilities are powerful for verifying the accuracy of your algebraic solutions.

Interval Notation

Interval notation is a succinct way of describing ranges of valid solutions. For the inequality \(10|1-x|<5\), once solved, we find the solution in terms of \(x\) to be between 0.5 and 1.5, inclusive. In interval notation, this is expressed as \([0.5, 1.5]\).
Here's how interval notation works:

• Brackets \([\cdot, \cdot]\) are used to include the endpoints of the interval, indicating that they are part of the solution.
• If endpoints were not included, parentheses \((\cdot, \cdot)\) would be used.

Using interval notation simplifies writing and reading solutions for inequalities, especially when dealing with large or complex numerical ranges.