Problem 65 - Calculus with Concepts in Calculus

Question

Solve the inequality. $$ |2 x+1| \geq 1 $$

Step-by-Step Solution

Answer

The solution is \(x \geq 0\) or \(x \leq -1\).

Step 1: Understanding the Absolute Value

The inequality involves an absolute value: \(|2x + 1| \geq 1\). This means that the expression inside can be either greater than or equal to 1, or less than or equal to -1. We must solve for both cases.

Step 2: Case 1: Positive Inequality

In this case, set the expression inside the absolute value greater than or equal to 1: \[2x + 1 \geq 1\] Subtract 1 from both sides: \[2x \geq 0\] Divide by 2: \[x \geq 0\] This gives us one part of the solution: \(x \geq 0\).

Step 3: Case 2: Negative Inequality

For the second case, set the expression inside the absolute value less than or equal to -1: \[2x + 1 \leq -1\] Subtract 1 from both sides: \[2x \leq -2\] Divide by 2: \[x \leq -1\] This gives us the other part of the solution: \(x \leq -1\).

Step 4: Combine the Solutions

The complete solution to the inequality is a combination of both cases. Therefore, the solution is:\[x \geq 0 \quad \text{or} \quad x \leq -1\] This represents all values of \(x\) that satisfy the original inequality.

Key Concepts

Absolute Value

When you encounter an absolute value in equations or inequalities, it represents the distance from zero on a number line. It is always non-negative. For example, in the expression \(|2x + 1| \geq 1\), the absolute value function takes whatever is inside the bars and considers both its positive and negative contributions.

The absolute value can be expressed in two separate scenarios: the expression inside is either greater than or equal to the given number or less than or equal to its negative.- For \(|2x + 1| \geq 1\), it means: - The expression is greater than or equal to 1. - The expression is less than or equal to -1.

This approach helps separate complex equations into simpler steps linearizing the problem.

Inequality Solutions

Solving inequalities is much like solving equations, but with some attention to detail, especially when involving absolute values or changing direction of inequalities. The key steps involve:- Isolate the term containing the variable.- Consider multiple scenarios that the inequality implies, especially when absolute values are present. - For example, \[2x + 1 \geq 1\] involves treating the expression without the absolute value directly. - Simultaneously, \[2x + 1 \leq -1\], considers the reversed inequality for negative scenarios.

Finally, remember when multiplying or dividing by negative numbers, the direction of the inequality sign should be reversed, but in this particular exercise, it isn't needed.

Piecewise Functions

Piecewise functions are a fundamental way of expressing solutions in parts, especially helpful in inequalities involving absolute values.

In our solution, combining the results, \[x \geq 0 \quad \ ext{or} \quad x \leq -1\], effectively uses the piecewise function approach.- This means that the function, or solution set, can change its rule, depending on the interval of interest.- It provides an elegant way to represent a comprehensive solution where multiple conditions satisfy an absolute inequality.

Understanding how to interpret and apply these intervals as separate solutions helps greatly in achieving an accurate answer.