# Absolute Value Equations Worksheet, Definition, Examples

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### Key Points about Absolute Value Equations

- Solving absolute value equations involves understanding the absolute value property and considering both the positive and negative values of the expression inside the absolute value bars.
- Graphing absolute value equations can be helpful in understanding their solutions and finding the solutions to the equation.
- Solving absolute value equations requires careful attention to detail and a methodical approach.

## Solving Absolute Value Equations

Solving absolute value equations is an important concept in mathematics that is used to solve equations that involve absolute values. Absolute value equations are equations in which the variable is within an absolute value expression. These equations can be tricky to solve, but with the right approach, anyone can master them.

To solve absolute value equations, one must first understand the absolute value property. The absolute value of a number is always positive, so when solving an absolute value equation, one must consider both the positive and negative values of the expression inside the absolute value bars. This can be done by setting up two equations and solving them separately.

Graphing absolute value equations can also be helpful in understanding their solutions. The graph of an absolute value equation is a V-shaped curve that opens upwards or downwards depending on the sign of the coefficient of the absolute value expression. By plotting the two cases of the equation separately, one can easily find the solutions to the equation.

**Common Core Standard:**

**How to Solve Absolute Value Equations**

Absolute value equations are equations that involve the absolute value of a variable. To solve these equations, follow these three steps:

**Isolate the Absolute Value**

The first step in solving an absolute value equation is to isolate the absolute value by moving all other terms to the other side of the equation. For example, consider the equation:

|2x – 3| = 5

To isolate the absolute value, add 3 to both sides:

|2x – 3| + 3 = 5 + 3

Simplify:

|2x – 3| = 8

**Split into Two Separate Equations**

The next step is to split the equation into two separate equations, one with a positive absolute value and one with a negative absolute value. For example, for the equation:

|2x – 3| = 8

We split it into two equations:

2x – 3 = 8 or 2x – 3 = -8

**Solve Each Equation Separately**

Finally, solve each equation separately. For the first equation, we have:

2x – 3 = 8

Add 3 to both sides:

2x = 11

Divide by 2:

x = 5.5

For the second equation, we have:

2x – 3 = -8

Add 3 to both sides:

2x = -5

Divide by 2:

x = -2.5

Therefore, the solutions to the original equation are x = 5.5 and x = -2.5.

In summary, to solve an absolute value equation, isolate the absolute value, split into two separate equations, and solve each equation separately.

**Absolute Value Equations Definition**

Absolute value equations are equations that involve the absolute value of a variable. The absolute value of a number is its distance from zero on the number line. It is always a non-negative value. For example, the absolute value of 5 is 5, and the absolute value of -5 is also 5.

In mathematical notation, the absolute value of a number is represented by two vertical bars surrounding the number. For instance, the absolute value of x is written as |x|.

An absolute value equation is an equation that contains an absolute value expression. The general form of an absolute value equation is:

|f(x)| = g(x)

where f(x) is a function of x, and g(x) is a function of x. The goal is to find the values of x that make the absolute value expression equal to g(x).

Absolute value equations can be solved using algebraic methods. The basic idea is to isolate the absolute value expression and then split the equation into two cases: one where the expression is positive and another where it is negative.

Overall, absolute value equations play an important role in mathematics and are used in various fields such as physics, engineering, and economics.

**Absolute Value Equations Examples**

Absolute value equations are equations that involve the absolute value of a variable. To solve these equations, one must isolate the absolute value expression and then consider both the positive and negative cases. Here are some examples of absolute value equations:

**Simple Absolute Value Equations**

A simple absolute value equation is an equation where the absolute value expression is the only one present. For example, consider the equation:

|2x + 1| = 5

To solve this equation, one must consider both the positive and negative cases. In the positive case, the equation becomes:

2x + 1 = 5

Solving for x, we get:

x = 2

In the negative case, the equation becomes:

-2x – 1 = 5

Solving for x, we get:

x = -3

Therefore, the solutions to the equation |2x + 1| = 5 are x = 2 and x = -3.

**Complex Absolute Value Equations**

A complex absolute value equation is an equation where there are other expressions present in addition to the absolute value expression. For example, consider the equation:

|2x – 1| + 3 = 7

To solve this equation, one must first isolate the absolute value expression. Subtracting 3 from both sides, we get:

|2x – 1| = 4

Next, we consider both the positive and negative cases. In the positive case, the equation becomes:

2x – 1 = 4

Solving for x, we get:

x = 2.5

In the negative case, the equation becomes:

-2x + 1 = 4

Solving for x, we get:

x = -1.5

Therefore, the solutions to the equation |2x – 1| + 3 = 7 are x = 2.5 and x = -1.5.

In summary, solving absolute value equations involves isolating the absolute value expression and considering both the positive and negative cases. Simple absolute value equations involve only the absolute value expression, while complex absolute value equations involve other expressions in addition to the absolute value expression.

**Graphing Absolute Value Equations**

Graphing absolute value equations is a fundamental skill in algebra that is essential for solving absolute value equations. The graph of an absolute value equation is a V-shaped graph that opens upwards or downwards, depending on the sign of the coefficient in front of the absolute value.

To graph an absolute value equation, you need to identify the vertex, the axis of symmetry, and the x and y intercepts. The vertex is the point where the V-shaped graph changes direction. The axis of symmetry is the line that passes through the vertex and divides the graph into two symmetrical halves. The x and y intercepts are the points where the graph intersects the x and y axes, respectively.

One way to graph an absolute value equation is to use a table of values. You can choose values of x and substitute them into the equation to find the corresponding values of y. Plotting these points on a coordinate plane will give you a rough sketch of the graph. To get a more accurate graph, you can plot more points or use a graphing calculator.

Another way to graph an absolute value equation is to use transformations. You can transform the graph of the basic absolute value function y = |x| by shifting it up or down, left or right, or reflecting it across the x or y axis. The general form of an absolute value equation is y = a|x – h| + k, where a, h, and k are constants that determine the shape and position of the graph.

In conclusion, graphing absolute value equations is an essential skill in algebra that requires knowledge of the basic properties of absolute value functions and the ability to use tables of values and transformations to graph them accurately. With practice, anyone can master this skill and solve absolute value equations with confidence.

**How to Write Absolute Value Equations**

To write an absolute value equation, the first step is to identify the expression that is within the absolute value bars. This expression can be a variable or a combination of variables and constants. Once you have identified the expression, you need to determine the condition that makes the absolute value of the expression equal to a given value.

For example, consider the equation |x – 2| = 5. The expression within the absolute value bars is x – 2. To determine the condition that makes the absolute value of x – 2 equal to 5, you need to consider two cases: when x – 2 is positive and when x – 2 is negative.

When x – 2 is positive, the equation becomes x – 2 = 5, which gives x = 7. When x – 2 is negative, the equation becomes -(x – 2) = 5, which simplifies to -x + 2 = 5, giving x = -3. Therefore, the solutions to the equation |x – 2| = 5 are x = -3 and x = 7.

Another example of writing an absolute value equation is |2x + 1| = 3. The expression within the absolute value bars is 2x + 1. To determine the condition that makes the absolute value of 2x + 1 equal to 3, you need to consider two cases: when 2x + 1 is positive and when 2x + 1 is negative.

When 2x + 1 is positive, the equation becomes 2x + 1 = 3, which gives x = 1. When 2x + 1 is negative, the equation becomes -(2x + 1) = 3, which simplifies to -2x – 1 = 3, giving x = -2. Therefore, the solutions to the equation |2x + 1| = 3 are x = -2 and x = 1.

In general, to write an absolute value equation, follow these steps:

- Identify the expression within the absolute value bars.
- Determine the condition that makes the absolute value of the expression equal to a given value.
- Consider two cases: when the expression is positive and when the expression is negative.
- Solve for the variable in each case.
- Write the solutions as a set.

By following these steps, you can write absolute value equations and find their solutions.

**Absolute Value Equations FAQ**

**What is an example of an absolute value equation?**

An absolute value equation is an equation that contains an absolute value expression. An example of an absolute value equation is |x – 3| = 5.

**What are the steps to solve absolute value equations?**

To solve an absolute value equation, follow these steps:

- Isolate the absolute value expression.
- Set up two equations, one with the expression equal to the positive value and one with the expression equal to the negative value.
- Solve both equations.
- Check the solutions by plugging them back into the original equation.

**How do you solve absolute value equations by graphing?**

To solve an absolute value equation by graphing, graph the left-hand side of the equation and the right-hand side of the equation on the same set of axes. The solution(s) will be the x-coordinate(s) of the point(s) where the two graphs intersect.

**What is the importance of solving absolute value equations?**

Solving absolute value equations is important in many areas of mathematics and science. It is used in solving optimization problems, in finding the distance between two points, and in solving problems involving absolute values.

**What are some common mistakes when solving absolute value equations?**

Some common mistakes when solving absolute value equations include:

- Forgetting to set up two equations, one with the expression equal to the positive value and one with the expression equal to the negative value.
- Incorrectly solving one or both of the equations.
- Forgetting to check the solutions by plugging them back into the original equation.

**What are some real-world applications of solving absolute value equations?**

Solving absolute value equations has many real-world applications, including:

- Finding the distance between two points on a coordinate plane.
- Solving problems involving temperature, such as finding the difference between the high and low temperatures in a day.
- Solving problems involving speed, such as finding the distance traveled by a car given its speed and time traveled.

**How do you check your solution when solving absolute value equations?**

To check your solution when solving an absolute value equation, plug the solution(s) back into the original equation and make sure that both sides of the equation are equal.

**What are the 3 steps to solving absolute value equations?**

The 3 steps to solving absolute value equations are:

- Isolate the absolute value expression.
- Set up two equations, one with the expression equal to the positive value and one with the expression equal to the negative value.
- Solve both equations.

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