Problem 60 - Calculus with Concepts in Calculus

Question

Solve the inequality. $$ |x-4|<0.1 $$

Step-by-Step Solution

Answer

\(3.9 < x < 4.1\)

Step 1: Understand the Problem

To solve the inequality \(|x-4| < 0.1\), we need to find the values of \(x\) that make the absolute value expression, which represents the distance between \(x\) and 4, less than 0.1.

Step 2: Remove the Absolute Value

The expression \(|x-4| < 0.1\) can be rewritten as two separate inequalities without the absolute value: \(x - 4 < 0.1\) and \(x - 4 > -0.1\). This is because the absolute value represents the positive distance, so it is bound between the negative and positive of any number (in this case, 0.1).

Step 3: Solve the First Inequality

Solve the inequality \(x - 4 < 0.1\) by adding 4 to both sides, resulting in \(x < 4.1\).

Step 4: Solve the Second Inequality

Similarly, solve \(x - 4 > -0.1\) by adding 4 to both sides. This gives \(x > 3.9\).

Step 5: Combine the Solutions

The solution to the original inequality \(|x-4| < 0.1\) is the intersection of the two inequalities: \(3.9 < x < 4.1\). This means \(x\) is greater than 3.9 and less than 4.1.

Key Concepts

Inequalities

Inequalities are mathematical expressions that show the relationship between two values when they are not exactly equal. An inequality can demonstrate if one value is less than, greater than, less than or equal to, or greater than or equal to another. Inequalities are denoted using symbols:

• "<" for less than
• ">" for greater than
• "≤" for less than or equal to
• "≥" for greater than or equal to

Inequalities are essential in mathematics for comparing numbers and expressing ranges of solutions, as is commonly seen in problems involving real-world contexts like cooking, budgeting, or construction. They help us understand not only specific solutions but the entire spectrum or interval of possibilities. Just like equations, inequalities can be solved to find sets of solutions that fulfill the condition described. They differ by considering a range instead of a specific value, which makes them quite versatile and widely applicable.

Solving Inequalities

Solving inequalities involves finding the set of all possible solutions that satisfy the inequality. The steps for solving an inequality often resemble those used for solving equations, but with a few key differences. Here's a simple guide:

• **Isolate the variable**: Utilize addition, subtraction, multiplication, or division to get the variable by itself on one side of the inequality. Be mindful when multiplying or dividing by negative numbers, as doing so reverses the inequality sign.
• **Simplify the expression**: Sometimes, it's necessary to combine like terms or use the distributive property to simplify one side or both sides of the inequality.
• **Test the solution**: Once you find a solution, or the range of solutions, plug values back into the original inequality to check for correctness.

For example, to solve \(x - 4 < 0.1\), you simply add 4 to both sides to obtain \(x < 4.1\). Similarly for \(x - 4 > -0.1\), you add 4 to obtain \(x > 3.9\). Combining these results gives \(3.9 < x < 4.1\). This solution describes an interval that comprises all values of \(x\) which are greater than 3.9 but less than 4.1.

Absolute Value

Understanding the concept of absolute value is crucial when dealing with inequalities that involve them. The absolute value of a number is its distance from zero on a number line, irrespective of direction. For instance, the absolute value of both \(-5\) and \(5\) is 5, as they are both five units away from zero.When solving inequalities that involve absolute values, such as \(|x-4| < 0.1\), the absolute value indicates a 'buffer' of space around the point, in this case, 4, within a radius less than 0.1 units. Thus, the solution begins by breaking the absolute value expression into a double inequality \(-0.1 < x-4 < 0.1\), capturing all possible values within that buffer zone.The presence of an absolute value means considering two scenarios: one capturing values below 4 by \(x-4 > -0.1\), and one capturing values above 4 by \(x-4 < 0.1\). Removing the absolute value is a dynamic way to transform these into straightforward linear inequalities, which are simpler to solve.