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Chapter 9: Problem 20

Find the solution set for each equation. $$7|3 x|+2=16$$

Short Answer

Expert verified
The solution set is {2/3, -2/3}

Step by step solution

01

Isolate the Absolute Value

In the first step, we isolate the absolute value expression. we can do so by subtracting 2 from both sides to get: \(7|3x| = 16 - 2 = 14\) and then we can divide by 7 to get: \(|3x| = 14 / 7 = 2\).
02

Split the Equation into Two Parts

Since the absolute function can provide two solutions (the positive and negative value), we split the equation into two equations: \(3x = 2\) and \(3x = -2\).
03

Solve Each Equation Individually

Now we will solve each equation in turn by dividing through by 3: \(x = 2 / 3\) from the first equation and \(x = -2 / 3\) from the second equation.

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Key Concepts

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

Isolate the Absolute Value
When we're faced with an absolute value equation like 7|3x|+2=16, our first step is to isolate the absolute value expression, which is |3x| in this case. This involves performing basic algebraic operations to get the absolute value by itself on one side of the equation. It's like carefully peeling away layers from an onion.

We start by subtracting 2 from both sides to peel off the initial layer, getting 7|3x| = 14. Then, we divide both sides by 7, unwrapping the last layer, which gives us |3x| = 2. Now the absolute value stands alone, and we're ready for the next step.
Split the Equation
With the absolute value isolated, think of the next step as preparing two separate paths for our equation. The property of absolute value is such that it equals both the positive and negative version of the same number鈥攎eaning that |3x|=2 could be resolved as 3x being either 2 or -2.

We turn our single absolute value equation into these two possibilities: 3x = 2 and 3x = -2. Splitting the equation in this way allows us to explore all potential solutions that satisfy the original equation.
Solve Each Equation Individually
Once we have our two separate equations鈥�3x = 2 and 3x = -2, we focus on solving each one individually as though they were their own puzzles. To solve for x, divide both sides of each equation by 3. This leaves us with x = 2 / 3 from the first and x = -2 / 3 from the second.

These are the two pieces of the solution set that answer our original equation. Always double-check by plugging these values back into the original equation to verify that both solutions indeed make the equation true.

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