How to solve an absolute value equation then find distance?

Absolute value equations can be challenging, but with the right approach, they can be solved effectively. Once you find the solution to the equation, you can use it to find the distance between two points on a number line. In this article, we will walk you through the steps to solve an absolute value equation and then find the distance. Let’s get started!

Solving an Absolute Value Equation

To solve an absolute value equation, follow these steps:

Step 1: Isolate the absolute value expression.

Step 2: Set up two equations, one with the positive form of the expression and the other with the negative form. Remove the absolute value bars in both equations.

Step 3: Solve each equation separately.

Step 4: Check your solutions by substituting them back into the original equation. Ensure that the left-hand side (LHS) and right-hand side (RHS) of the equation match.

Step 5: If the solutions of both equations are correct, your answer is the union of these solutions.

How to solve an absolute value equation then find distance?

The distance between any two points on a number line can be found by taking the absolute value of the difference between those points.

For example, if two points are given as A and B, the distance between them (d) is given by the formula:
d = |A – B|

Let’s go through a step-by-step example to make it clearer.

Example: Solve the absolute value equation |2x + 3| = 7 and find the distance.

Step 1: Isolate the absolute value expression:
2x + 3 = 7 or 2x + 3 = -7

Step 2: Set up two equations:
2x + 3 = 7
2x + 3 = -7

Step 3: Solve each equation separately:
For the first equation, we have:
2x = 4
x = 2

For the second equation, we have:
2x = -10
x = -5

Step 4: Check your solutions:
Substituting x = 2 into the original equation |2x + 3| = 7:
|2(2) + 3| = 7
|4 + 3| = 7
|7| = 7
7 = 7 (LHS = RHS)

Substituting x = -5 into the original equation |2x + 3| = 7:
|2(-5) + 3| = 7
|-10 + 3| = 7
|-7| = 7
7 = 7 (LHS = RHS)

Both solutions are correct.

Step 5: Answer
The solutions to the equation |2x + 3| = 7 are x = 2 and x = -5. The distance between these two points is:
d = |2 – (-5)|
d = 7

Therefore, the distance between the two points is 7 units.

Frequently Asked Questions (FAQs)

Q1: Can an absolute value equation have more than two solutions?

A1: Yes, an absolute value equation can have zero, one, or two solutions depending on the equation.

Q2: Are there any shortcuts to solve absolute value equations?

A2: There are no specific shortcuts, but understanding the steps and practicing can make it easier.

Q3: Can an absolute value equation have no solutions?

A3: Yes, it is possible for an absolute value equation to have no solutions if the equation is contradictory.

Q4: How do absolute value equations relate to distance?

A4: Absolute value equations help us find the distance between two points on a number line.

Q5: Can an absolute value equation have an infinite number of solutions?

A5: No, absolute value equations cannot have an infinite number of solutions. They either have zero, one, or two solutions.

Q6: Can we solve absolute value equations using graphing?

A6: Yes, graphing can also be used to solve absolute value equations. The solutions are the x-values where the graph intersects with the x-axis.

Q7: Can an absolute value equation involve variables other than x?

A7: Yes, absolute value equations can involve any variable, not just x. The solution will be in terms of the variable given in the equation.

Q8: Do we need to consider different cases when solving absolute value equations?

A8: Yes, we need to consider different cases when setting up the equations, one for the positive form and one for the negative form, to ensure we capture all possible solutions.

Q9: Are there any real-life applications of absolute value equations?

A9: Yes, absolute value equations are used in various fields such as physics, engineering, economics, and computer science to solve problems related to distance, optimization, inequalities, and more.

Q10: Can we solve absolute value equations using inequalities?

A10: Yes, absolute value equations can be solved using inequalities, and the solutions will include the values that satisfy the inequality.

Q11: Can we solve an absolute value equation by just removing the absolute value bars?

A11: No, we cannot simply remove the absolute value bars. We need to set up two separate equations, considering both the positive and negative forms of the expression.

Q12: Can an absolute value equation have non-integer solutions?

A12: Yes, absolute value equations can have non-integer solutions. The solutions can be rational, irrational, or even complex numbers depending on the given equation.

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