Problem 48 - Beginning and Intermediate Algebra

Question

Write an absolute value equation that has a solution set of \(\\{-1.4,1.4\\}\)

Step-by-Step Solution

Answer

The absolute value equation with a solution set of \(\{-1.4, 1.4\}\) is \(|x - 1.4| = 2.8\).

Step 1: Understand properties of absolute value equations

Absolute value equations have the form \(|ax+b|=c\), where a, b, and c are constants. The solutions of these equations are often given as pairs of values, representing the positive and negative values of the result.

Step 2: Determine the expression inside the absolute value brackets

Since the solutions given are -1.4 and 1.4, they must be the result of the expression inside the absolute value brackets when the equation is satisfied. To find the relationship between the solutions, we can compute their average: \[\frac{-1.4 + 1.4}{2} = 0.\] The value, 0, indicates that the expression inside the absolute value brackets must be symmetric around the origin, which leads us to the expression \(x + 1.4\) and \(x - 1.4\).

Step 3: Determine the constant outside the absolute value brackets

Now that we have an expression inside the absolute value brackets, we need to find the constant that would provide us with the other solution of the equation. We'll start with the expression \(x - 1.4\) inside the absolute value brackets and try to obtain the solution of -1.4: \[|-1.4 - 1.4| = 2.8.\] Since we want the absolute value equation to provide us with the given solutions, we need the constant outside the absolute value brackets to be equal to 2.8.

Step 4: Write the absolute value equation

With the information gathered in Steps 2 and 3, we can now write the absolute value equation: \[|x - 1.4| = 2.8\] This absolute value equation has the solution set of {-1.4, 1.4}.

Key Concepts

Solution Set

The solution set of an absolute value equation is the collection of all values that satisfy the equation. For our given example, the solution set is \(-1.4, 1.4\). This means that these are the values of \(x\) that make the absolute value equation true. In simple terms, when we substitute these values back into the equation, they will result in a true statement.
To find a solution set, you first need to solve the absolute value equation, typically represented as \( |ax + b| = c \).
Here are the basic steps to find the solution set:

• Remove the absolute value by considering both the positive and negative scenarios that satisfy the equation.
• Solve the resulting simple linear equations.
• The solutions to these equations make up the solution set.

Keep in mind that the solution set can have two solutions, as shown in our exercise where both \(-1.4\) and \(1.4\) are valid solutions.

Properties of Absolute Value

Understanding the properties of absolute value is crucial when solving these equations. The absolute value of a number is its distance from zero on a number line, regardless of direction. This is why absolute value is always non-negative.
Some important properties of absolute value include:

• \( |x| = x \) if \( x \geq 0 \), otherwise \( |x| = -x \) if \( x < 0 \).
• The equation \( |a| = b \) has solutions \( a = b \) and \( a = -b \) if \( b \geq 0 \).
• When creating an absolute value equation, recognize that it always reflects situations with two opposite results, linked inherently to the nature of distance.

These properties help predict the behavior of the absolute value equations, which is why they often yield pairs of solutions.

Symmetric Solutions

Symmetric solutions in absolute value equations refer to the property that solutions are often equal in magnitude but opposite in sign. This happens because absolute value measures distance from a point, treating direction as irrelevant. In our exercise, we have the solutions \(-1.4\) and \(1.4\), which clearly show this symmetry.
Here’s why it works this way:

• The expression inside the absolute value must be equidistant from zero, explaining the equal but opposite nature of solutions.
• This symmetry is a consistent feature of absolute value equations, highlighting their predictable result of two solutions if \( |x| = c \) where \( c > 0 \).

These symmetric solutions make it easier to understand and solve absolute value equations, as you can anticipate this behavior when analyzing the problem.