Problem 33 - Algebra for College Students

Question

Solve each equation and inequality. \(|1-2 x|<2\)

Step-by-Step Solution

Answer

\(-\frac{1}{2} < x < \frac{3}{2}\)

Step 1: Identify Absolute Value Inequality

The given inequality is \(|1-2x| < 2\). This is an absolute value inequality. The goal is to find the range of values for \(x\) that satisfy this inequality.

Step 2: Rewrite as Two Linear Inequalities

For an inequality of the form \(|a| < b\), we can rewrite it as two separate linear inequalities: \(a < b\) and \(a > -b\). Applying this to our problem, we get:1. \(1 - 2x < 2\)2. \(1 - 2x > -2\).

Step 3: Solve First Linear Inequality

Let's solve \(1 - 2x < 2\):- Subtract 1 from both sides: \(-2x < 1\)- Divide both sides by \(-2\). Remember, dividing by a negative number reverses the inequality sign: \(x > -\frac{1}{2}\).

Step 4: Solve Second Linear Inequality

Now solve \(1 - 2x > -2\):- Subtract 1 from both sides: \(-2x > -3\)- Divide both sides by \(-2\). Again, reverse the inequality sign: \(x < \frac{3}{2}\).

Step 5: Combine Solutions

The solution to the inequalities \(x > -\frac{1}{2}\) and \(x < \frac{3}{2}\) must be considered together. This results in the combined inequality:- \(-\frac{1}{2} < x < \frac{3}{2}\).

Key Concepts

Linear Inequalities

Linear inequalities are like linear equations, but instead of using an equal sign, they use inequality symbols like >, <, ≥, or ≤. These symbols indicate a range of possible values rather than a single solution. For instance, if we have the inequality \(3x + 5 < 11\), we need to think about the range of x-values that will make this true. We aim to isolate \(x\) on one side of the inequality to find these values. To solve them, you follow similar steps to solving linear equations:

• First, perform operations to isolate the variable on one side as you would in a regular equation.
• If you multiply or divide by a negative number, remember to flip the inequality sign. This is a crucial part of working with inequalities.
• Finally, express your solution as a range, such as \(x < 2\).

Understanding linear inequalities helps to work effectively with expressions where the solutions need to represent a range rather than a unique value.

Solving Equations

Solving equations involves finding the value of the unknown variable that makes the equation true. In the context of absolute value inequalities like \(|1-2x| < 2\), solving equations is a two-step process. The absolute value expression separates the inequality into two parts:

• The first part where the expression inside the absolute value is less than the number: \(1 - 2x < 2\).
• The second part where the expression inside is greater than the negative of that number: \(1 - 2x > -2\).

Once split, each part becomes a separate equation that you solve individually:

• For \(1 - 2x < 2\): Subtract 1 from each side to get \(-2x < 1\), then divide by \(-2\) flipping the inequality sign to get \(x > -\frac{1}{2}\).
• For \(1 - 2x > -2\): Similarly, subtract 1 to achieve \(-2x > -3\), and divide by \(-2\) to reverse the inequality, leading to \(x < \frac{3}{2}\).

The process of solving these separate equations reveals the solution which must satisfy both conditions. Remember, the operations are the same, but the outcome helps define a range.

Inequality Solutions

Inequality solutions define a set of possible values for the variable, often expressed as a range. In our example of solving \(|1-2x| < 2\), we combine the solutions of our split equations to form a single range of values for \(x\):

• The solution \(x > -\frac{1}{2}\) means \(x\) must be greater than \(-\frac{1}{2}\).
• The solution \(x < \frac{3}{2}\) indicates \(x\) must also be less than \(\frac{3}{2}\).

When combining these solutions, we arrive at the inequality \(-\frac{1}{2} < x < \frac{3}{2}\), revealing the values \(x\) can take. This type of solution is significant because it gives us a range rather than a single number. Visualizing this on a number line helps make the concept clear: imagine shading the section between \(-\frac{1}{2}\) and \(\frac{3}{2}\), excluding endpoints if it's a strict inequality. Always be mindful of whether the inequality includes \(<\) or \(≤\) signs, as this affects the interpretation of the range boundaries.