Problem 36 - College Algebra

Question

For the following exercises, solve the inequality. If possible, find all values of \(a\) such that there are no \(x\) -intercepts for \(f(x)=2|x+1|+a\)

Step-by-Step Solution

Answer

For \(a > 0\), there are no \(x\)-intercepts.

Step 1: Understand the Problem

We need to find all values of \(a\) such that the function \(f(x) = 2|x+1| + a\) has no \(x\)-intercepts. This means \(f(x)\) should never equal zero.

Step 2: Set the Function Equals to Zero

To find x-intercepts, set \(f(x) = 0\). Thus, we have \(2|x+1| + a = 0\). Simplify to get \(|x+1| = -\frac{a}{2}\).

Step 3: Analyze Absolute Value Condition

The expression \(|x+1|\) denotes absolute value, which is always non-negative. Therefore, \(-\frac{a}{2}\) must be negative for \(|x+1| = -\frac{a}{2}\) to have no solutions.

Step 4: Solve Inequality for a

Set \(-\frac{a}{2} < 0\) to ensure \(|x+1|\) never equals \(-\frac{a}{2}\). This simplifies to \(a > 0\).

Step 5: Conclusion

If \(a > 0\), the function \(f(x) = 2|x+1| + a\) has no \(x\)-intercepts because the absolute value cannot be negative.

Key Concepts

Understanding Absolute Value

The concept of absolute value is fundamental in mathematics. It represents the non-negative value of a number or expression, regardless of its sign. Let's break it down:

• For any real number, the absolute value is its distance from zero on the number line.
• It is always non-negative, meaning it cannot be less than zero.
• For example, \(|x+1|\) is the absolute value of \(x+1\), which is always zero or positive.

When analyzing inequalities, understanding that the absolute value cannot be negative helps us determine the conditions needed for other parts of the function or equation to not be satisfied. In our problem, recognizing \(|x+1|\) cannot equal a negative number is key to solving the inequality.

Finding X-Intercepts

X-intercepts of a function occur where the value of the function is zero. These are the points on the graph where it crosses the x-axis. Here's a quick overview to identify them:

• Set the function equal to zero: this represents the line where the function touches or crosses the x-axis.
• Solve for the variable, typically \(x\), to find the points of intersection.

For the function given, \(f(x) = 2|x+1| + a\), setting the function equal to zero allows us to determine the conditions under which there are no x-intercepts, i.e., \(2|x+1| + a = 0\). By solving this setup, we discovered that \(a\) must be greater than zero to ensure the function never intersects the x-axis.

Function Analysis

When analyzing a function like \(f(x) = 2|x+1| + a\), we explore how it behaves under various conditions.

• First, identify and understand each part of the function.
• The absolute value part, \(2|x+1|\), dominates the shape and position of the graph.
• Next, consider how the addition of \(a\) affects the graph.

The constant \(a\) shifts the entire graph up or down, depending on its value. Studying \(2|x+1|\), the graph typically forms a V-shape, with the vertex pointing upward if \(a > 0\). The factor of two affects how steep the V is. If the condition for \(a > 0\) is met, the vertex always sits above the x-axis, hence, no intersection with the x-axis.

Inequality Conditions

Inequalities such as \(-\frac{a}{2} < 0\) require careful analysis. They help determine valid conditions that make the inequality true. Here's how to approach them:

• Identify the expressions to isolate the variable of interest.
• Determine the algebraic steps needed to solve the inequality.
• Consider the implications of the inequality for the other elements of the function.

By solving \(-\frac{a}{2} < 0\), we determine that \(a > 0\) is necessary to ensure no x-intercepts occur. Hence, interpreting inequality conditions helps us find the solution that meets the criteria posed by the problem. These solutions preserve the characteristics of the absolute value component of the function, confirming no x-intercepts exist when \(a > 0\).