Solve the Absolute Value Equation: Find X in |2x + 6| = 1

Q: Why does an absolute value equation have two solutions?

Think of absolute value as distance from zero. If something is 1 unit away from zero, it could be at +1 or -1. Similarly, $ |2x + 6| = 1 $ means the expression inside could equal +1 or -1.

Q: How do I know which equation to set up?

Always set up both cases: make the inside expression equal to the positive value and the negative value. For $ |2x + 6| = 1 $, that's $ 2x + 6 = 1 $ and $ 2x + 6 = -1 $.

Q: What if I only get one solution when I solve?

You should always get two solutions unless the absolute value equals zero. Double-check your arithmetic - you might have made a calculation error in one of the equations.

Q: Do I need to check both answers?

Yes! Always substitute both solutions back into the original equation. This catches any arithmetic mistakes and confirms that both values actually work.

Q: Can absolute value equations have no solutions?

Yes, if the absolute value equals a negative number. Since absolute values are always non-negative, equations like $ |x| = -3 $ have no solutions.

Absolute Value Equations with Two Solutions

$\left|2x+6\right|=1$

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Understand the problem

$\left|2x+6\right|=1$

Step-by-step solution

To solve the absolute value equation $|2x + 6| = 1$ , we must consider the definition of absolute value.

Step 1: Set up two separate equations corresponding to the positive and negative scenarios:

Equation 1: $2x + 6 = 1$
Equation 2: $2x + 6 = -1$

Step 2: Solve each equation independently for $x$ .

Solving Equation 1:
Start with $2x + 6 = 1$ .
Subtract 6 from both sides: $2x = 1 - 6$ .
Thus, $2x = -5$ .
Divide both sides by 2: $x = -\frac{5}{2}$ or $x = -2.5$ .

Solving Equation 2:
Start with $2x + 6 = -1$ .
Subtract 6 from both sides: $2x = -1 - 6$ .
Thus, $2x = -7$ .
Divide both sides by 2: $x = -\frac{7}{2}$ or $x = -3.5$ .

Combining both results, the solutions to the equation $|2x + 6| = 1$ are:

$x = -2.5$ and $x = -3.5$ .

Final Answer

$x=-2.5$ , $x=-3.5$

Key Points to Remember

Essential concepts to master this topic

Definition: Absolute value equals distance, creating two possible cases
Method: Set up $2x + 6 = 1$ and $2x + 6 = -1$
Verify: Check both solutions: $|2(-2.5) + 6| = 1$ and $|2(-3.5) + 6| = 1$ ✓

Common Mistakes

Avoid these frequent errors

Solving only one equation instead of both cases
Don't solve just $2x + 6 = 1$ and stop = you miss half the solutions! Absolute value creates two scenarios because distance can be measured in either direction. Always set up both the positive and negative cases to find all solutions.

Practice Quiz

Test your knowledge with interactive questions

$\left|x\right|=5$

FAQ

Everything you need to know about this question

Why does an absolute value equation have two solutions?

Think of absolute value as distance from zero. If something is 1 unit away from zero, it could be at +1 or -1. Similarly, $|2x + 6| = 1$ means the expression inside could equal +1 or -1.

How do I know which equation to set up?

Always set up both cases: make the inside expression equal to the positive value and the negative value. For $|2x + 6| = 1$ , that's $2x + 6 = 1$ and $2x + 6 = -1$ .

What if I only get one solution when I solve?

You should always get two solutions unless the absolute value equals zero. Double-check your arithmetic - you might have made a calculation error in one of the equations.

Do I need to check both answers?

Yes! Always substitute both solutions back into the original equation. This catches any arithmetic mistakes and confirms that both values actually work.

Can absolute value equations have no solutions?

Yes, if the absolute value equals a negative number. Since absolute values are always non-negative, equations like $|x| = -3$ have no solutions.

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