# Absolute Value Word Problems Worksheets

### What Are Absolute Value Word Problems?

Absolute value word problems involve using the concept of absolute value to solve real-life situations or scenarios. Absolute value represents the distance of a number from zero on a number line, regardless of its direction. In word problems, absolute value is often used to represent a magnitude or a distance, and it is denoted by vertical bars (| |).

Absolute value word problems can cover various topics and contexts, such as distances, temperatures, time intervals, financial transactions, and more. These problems typically require determining the absolute value of a quantity or comparing absolute values to find a solution.

For example, here’s a simple absolute value word problem:

**Problem:** A football team gains or loses yardage on each play. On one play, the team loses 12 yards. On the next play, they gain 8 yards. What is the total yardage change?

**Solution:** To find the total yardage change, we need to calculate the absolute value of the sum of the yardage gained and lost. In this case, the team lost 12 yards and gained 8 yards. So, the total yardage change is |(-12) + 8| = |-4| = 4 yards. Therefore, the total yardage change is 4 yards.

Absolute value word problems can become more complex, involving multiple quantities, comparisons, or inequalities. However, the underlying principle remains the same—finding the absolute value of a quantity to represent a magnitude or distance in the problem context.

### How to Solve These Types of Problems

Solving absolute value word problems involves understanding the concept of absolute value and applying it to real-life scenarios. Here’s a step-by-step guide to help you solve absolute value word problems:

Begin by thoroughly understanding the problem statement. Identify the information provided and what you’re being asked to find. Pay attention to keywords and phrases that indicate the use of absolute value, such as “distance,” “deviation,” or “magnitude.”

**Identify the absolute value expression** – Look for the specific quantity or expression within the problem that is enclosed within absolute value symbols (| |). This expression represents the magnitude or distance from zero.

**Set up the equation** – Create an equation based on the information provided in the problem. Consider two scenarios – one where the absolute value expression is positive and another where it is negative. Set up the equation accordingly.

**Solve for both cases** – Solve the equation for each case you identified in the previous step. Remove the absolute value symbols and isolate the variable to find its possible values.

**Check your solutions** – Substitute the potential solutions back into the original problem and evaluate if they satisfy the given conditions. For absolute value problems involving distances, ensure that the solutions make sense within the context of the problem. State the final solution or solutions based on the context of the problem. Depending on the problem, you may need to provide a single answer, a range of values, or a set of possible solutions.

**Check for extraneous solutions** – Sometimes, when solving absolute value equations, extraneous solutions may arise. These are solutions that do not satisfy the original problem conditions. Double-check your solutions and ensure they are valid.

### This Skill In The Real World

Here are a few examples of situations where absolute value is used:

**Distance and Displacement** – Absolute value is commonly used to represent distances or displacements. For instance, if you are calculating the distance between two cities or the displacement of an object from its initial position, you would use absolute value to ensure the result is positive.

**Temperature** – Absolute value can be used in temperature-related problems. For example, if you need to determine the difference in temperature between two points, you would take the absolute value to obtain a positive value regardless of whether one temperature is higher or lower than the other.

**Finance and Economics** – Absolute value is relevant in finance and economics when dealing with measures of deviation or error. For instance, calculating the absolute value of the difference between an actual value and an expected value can help determine the magnitude of the deviation or error.

**Optimization and Constraints** – In optimization problems, where you aim to maximize or minimize a certain quantity, absolute value can be used to express constraints. For example, if you have a constraint that the difference between two variables must be less than a certain value, you would use absolute value to formulate the constraint.

**Physics and Science –** Absolute value is frequently used in physics and other sciences. It can represent quantities like velocity, acceleration, force, or electric charge, where direction may not be relevant, and only the magnitude matters.

### What Types of Jobs and Careers Use This Skill?

Several professions and fields require regularly solving absolute value word problems as part of their job responsibilities. Here are a few careers where the use of absolute value is common:

**Mathematics and Statistics**– Mathematicians and statisticians often encounter absolute value word problems in their work. They use absolute value to analyze data, measure distances, calculate deviations, and solve various mathematical equations and inequalities.**Physics and Engineering**– Professionals in physics and engineering regularly work with absolute value concepts. They apply absolute value when dealing with measurements, vectors, forces, distances, and deviations in various physical phenomena.**Economics and Finance**– Economists and financial analysts frequently use absolute value in their work. They apply it to analyze deviations, calculate errors, measure differences, and determine the magnitude of changes in economic indicators, financial data, or investment returns.**Operations Research and Optimization**– Professionals in operations research and optimization use absolute value to formulate constraints and objective functions in mathematical models. They often encounter absolute value word problems when optimizing processes, resource allocation, scheduling, or decision-making.**Data Analysis and Machine Learning**– Data analysts and machine learning specialists encounter absolute value problems when working with datasets. They use absolute value to calculate error metrics, evaluate model performance, handle outliers, or measure distances between data points.**Geography and Navigation**– Geographers, cartographers, and navigation experts may use absolute value when calculating distances between locations or analyzing spatial data. Absolute value helps measure the magnitude of differences or deviations in geographic coordinates.**Transportation and Logistics**– Professionals in transportation and logistics rely on absolute value to solve problems related to routes, distances, and delivery times. They may use it to minimize deviations, calculate optimal paths, or evaluate the efficiency of transportation networks.