Problem 69 - Calculus with Concepts in Calculus

Question

Solve the inequality. $$ -1<|4-2 x|<1 $$

Step-by-Step Solution

Answer

The solution is \(\frac{3}{2} < x < \frac{5}{2}\).

Step 1

Identify the compound inequality. The given inequality is \(-1 < |4 - 2x| < 1\). The absolute value inequality \(|A| < B\) means \(-B < A < B\). Here, \(|4 - 2x| < 1\), we interpret this as two inequalities: \(-1 < 4 - 2x\) and \(4 - 2x < 1\).

Step 2

Solve the first inequality \(-1 < 4 - 2x\).Subtract 4 from both sides to isolate the \(-2x\) term:\[-1 - 4 < -2x \-5 < -2x\] Now, divide both sides by \(-2\) to solve for \(x\), and remember to reverse the inequality sign:\[x < rac{5}{2}\]

Step 3

Solve the second inequality \(4 - 2x < 1\).Subtract 4 from both sides:\[4 - 4 - 2x < 1 - 4 \-2x < -3\]Divide by \(-2\), and reverse the inequality sign:\[x > rac{3}{2}\]

Step 4

Combine the solutions from both inequalities.From Steps 2 and 3, we have two results:\[ x < rac{5}{2} \] and \[ x > rac{3}{2} \].These can be combined to form the solution:\[rac{3}{2} < x < rac{5}{2}\]

Key Concepts

Understanding Absolute Value Inequalities

Absolute value inequalities are an extension of the concept of absolute values, which measure the distance of a number from zero on the number line. For example, the absolute value of a number \(|x|\) can be interpreted as "the distance of \(x\) from zero."

When dealing with absolute value inequalities, such as \(|A| < B\), we are interested in the range of values that the inside expression \(A\) can take. This translates to setting up a compound inequality:

• \(-B < A\)
• \(A < B\)

These inequalities tell us the extent to which \(A\) can deviate from zero without exceeding the threshold \(B\). In the example \(-1 < |4 - 2x| < 1\), the absolute value dictates two separate situations:

• \(-1 < 4 - 2x\)
• \(4 - 2x < 1\)

These two inequalities help us identify the boundary conditions for solving the inequality.

Breaking Down Compound Inequalities

Compound inequalities are sets of two or more inequalities joined together by either an "and" or an "or". In the context of the exercise, we have an "and" compounded inequality, indicated by the interval notation \( -1 < |4 - 2x| < 1 \).

To solve such compound inequalities, we work with each component separately. In our case, this leads us to solve the following two inequalities:

• \(-1 < 4 - 2x\)
• \(4 - 2x < 1\)

Both parts need to be true simultaneously for the original compound inequality to hold true. The solution to these inequalities is then combined to form a continuous range of possible solutions that satisfy the original statement.

This logical "and" condition enables us to determine a specific range for \(x\), forming the final solution interval: \(\frac{3}{2} < x < \frac{5}{2}\). This solution captures all values that make both components true at the same time.

Mastering Algebraic Manipulation

Algebraic manipulation involves various operations performed to simplify or solve equations and inequalities. Mastery of these techniques is critical in dealing with mathematical problems efficiently.

In the current inequality problem, manipulating the equations consists of isolating the variable \(x\). For instance, consider the step:

• For \(-1 < 4 - 2x\), subtract 4 from both sides to get \(-5 < -2x\). Dividing by \(-2\), and reversing the sign, gives \(x < \frac{5}{2}\).
• For \(4 - 2x < 1\), subtracting 4 from both sides yields \(-2x < -3\). Dividing by \(-2\) and reversing the inequality sign results in \(x > \frac{3}{2}\).

Key operations include:

• Adding or subtracting numbers to both sides
• Multiplying or dividing both sides by a negative number, which requires flipping the inequality sign
• Combining and simplifying expressions

Through systematic application of these algebraic manipulations, the solution emerges seamlessly, leading to a neatly defined range for \(x\). This ensures that students not only find the solution but also fully understand the process and principles involved.