Problem 2 - Algebra for College Students

Question

Solve each inequality and graph the solutions. \(|x|<1\)

Step-by-Step Solution

Answer

The solution is \(-1 < x < 1\) and is graphed by an open interval between -1 and 1 on the number line.

Step 1: Understand the Absolute Value Inequality

The inequality \(|x| < 1\) means that the absolute value of \(x\) is less than 1. This implies \(x\) is within 1 unit of 0 on the number line.

Step 2: Break Down the Absolute Value Inequality

The inequality \(|x| < 1\) translates to two separate inequalities: \(-1 < x < 1\). This means \(x\) must be greater than \(-1\) and less than \(1\).

Step 3: Graph the Solution

To graph \(-1 < x < 1\): Draw a number line, place open circles at \(-1\) and \(1\) (to indicate these values are not included), and shade the region between them to indicate all numbers between \(-1\) and \(1\) are solutions.

Key Concepts

Graphing Inequalities

Graphing inequalities is an essential skill for visually understanding the range of solutions. When dealing with inequalities involving absolute values, such as \(\|x\| < 1\), the approach starts with breaking down the inequality into its linear components.
Follow these steps when graphing an inequality:

• Solve the inequality as an equation first (e.g., solve \(-1 < x < 1\)).
• Determine the endpoints, which, for \(\|x\| < 1\), are \(-1\) and \(1\).
• On a number line, use open circles at the endpoints to symbolize that the points themselves are not included in the solution set.
• Shade the area between these circles. The shaded area indicates all the values \(x\) can take.

This visual representation helps in understanding not just the set of solutions but also the breadth of possible values \(x\) can adopt.

Number Line Representation

The number line offers a straightforward way to visualize solution sets for inequalities like \(\|x\| < 1\). This involves a simple mapping of values and their range:

To place our inequality on a number line:

• Start by drawing a horizontal line and marking key numbers, like \(-1\) and \(1\), on it.
• Since \(-1 < x < 1\) involves open intervals, use open circles at these numbers to illustrate that \(x\) cannot be exactly \(-1\) or \(1\).
• Fill in the region between these two circles to represent that all values between \(-1\) and \(1\) satisfy the inequality.

Such a representation is crucial, especially as it forms the basis for more complicated math concepts. It helps in intuitively understanding both the range and limitations of the solutions.

Multi-step Inequality Solving

Solving multi-step inequalities involves evaluating and rewriting inequalities in their simplest form. This process is crucial for clear and accurate graphing. Consider our example \(\|x\| < 1\). Here, we break it into simpler expressions:

Here’s how you solve it step by step:

• Recognize that \(\|x\| < 1\) splits into \(-1 < x < 1\), which is a compound inequality due to the absolute value.
• Treat the two simultaneous inequalities separately: begin with \(-1 < x\) and \(x < 1\), and view them as interconnected.
• Check for any simplification or further steps (like dividing by a positive number) needed only if other variables or constants are involved.
• Once simplified, the multi-step approach should make the graphing process straightforward, as each step directly contributes to constructing the number line solution efficiently.

Mastering multi-step inequality solving equips you with versatility when dealing with a variety of algebraic equations.