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Chapter 1: Problem 17

Find the real solutions, if any, of each equation. $$ |-2| x=4 $$

Short Answer

Expert verified
The solution is \(x = 2\).

Step by step solution

01

Simplify the absolute value

First, determine the value of the absolute value term. Since \(|-2| = 2\), the equation can be simplified.
02

Rewrite the equation

Replace \(|-2|\) with 2 in the equation, which gives us: \[2x = 4\]
03

Solve for x

Divide both sides of the equation by 2 to isolate \(x\): \[x = \frac{4}{2} \] Simplifying this gives \[x = 2\].
04

Verify the solution

Substitute \(x = 2\) back into the original equation to verify: \(|-2| x = 4\) becomes \( 2 \times 2 = 4\). Since this is true, \(x = 2\) is a valid solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Absolute Value
The absolute value of a number refers to its distance from zero on the number line. It is always a non-negative number. For any real number, let's call it **a**, the absolute value is denoted as \(|a|\).
The absolute value function is defined as:
  • If **a** is positive or zero, then \(|a| = a\).
  • If **a** is negative, then \(|a| = -a\).
In our exercise, we have \(|-2|\), which means we need the non-negative value of -2. Since -2 is negative, by definition we take its positive counterpart, making \(|-2| = 2\). This simplification is crucial to solving the equation easily.
Solving Equations
Solving equations involves finding the value(s) of the variable(s) that make the equation true. Here's a step-by-step guide:
  • Identify the type of equation: In the given example, \( |-2| x = 4 \), we deal with an absolute value and a basic linear equation.
  • Simplify the absolute value: As discussed, \(|-2| = 2\), so rewrite the equation: \(2x = 4\).
  • Isolate the variable: To solve for **x**, we need to get **x** by itself. This is done by dividing both sides by 2, leading to: \(x = \frac{4}{2} \).
  • Simplify the expression: Finally, \( x = 2 \), which is the solution to the equation.
By breaking down the problem into manageable steps, solving equations becomes much more straightforward.
Verification of Solutions
Verification is an important step in solving equations because it confirms that our solution is correct. Here's how to verify solutions:
  • Re-substitute the solution: Take the solution you found, in this case, \( x = 2 \) and plug it back into the original equation. Our equation is \( |-2| x = 4 \). When we substitute **x** with 2, it becomes \(2 \times 2 = 4 \).
  • Simplify and check equality: Simplify both sides of the equation. If they are equal, the solution is correct. Here, we get \(4 = 4 \), which holds true.
Verifying not only ensures the accuracy of the solution but also reinforces the understanding of each step involved in solving the equation.

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Most popular questions from this chapter

Make up an inequality that has no solution. Make up an inequality that has exactly one solution.

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