Problem 36 - College Algebra with Corequisite Support

Question

For the following exercises, solve the equation involving absolute value. $$ -|2 x+1|=-3 $$

Step-by-Step Solution

Answer

The solutions for the equation are \(x = 1\) and \(x = -2\).

Step 1: Recognize the Absolute Value Property

Recall that the absolute value function \(|x|\) represents the distance of \(x\) from zero on the number line, which means it is always non-negative. In this equation, we have \(-|2x + 1| = -3\). The negative sign outside means that the absolute value expression itself equals \(3\). So, we get \(|2x + 1| = 3\).

Step 2: Set Up the Equations

Since \(|2x + 1| = 3\), this means that \(2x + 1\) could be either \(3\) or \(-3\) because absolute value measures distance without regard to direction. Thus, we set up two separate equations: \(2x + 1 = 3\) and \(2x + 1 = -3\).

Step 3: Solve the First Equation

Solve \(2x + 1 = 3\). First, subtract \(1\) from both sides to get \(2x = 2\). Then, divide both sides by \(2\) to find \(x = 1\).

Step 4: Solve the Second Equation

Solve \(2x + 1 = -3\). Start by subtracting \(1\) from both sides to get \(2x = -4\). Next, divide both sides by \(2\) to find \(x = -2\).

Step 5: Verify the Solutions

Substitute \(x = 1\) and \(x = -2\) back into the original equation \(-|2x + 1| = -3\) to ensure both values satisfy the equation.For \(x = 1\): \(-|2(1) + 1| = -|2 + 1| = -3\), which satisfies the equation.For \(x = -2\): \(-|2(-2) + 1| = -|-4 + 1| = -3\), which also satisfies the equation. Both solutions are verified.

Key Concepts

Properties of Absolute Value

When solving equations with absolute value, it's crucial to understand what absolute value means. The absolute value of a number, represented as \(|x|\), is the number's distance from zero on the number line. This means it is always non-negative.

For example,

• \(|5| = 5\)
• \(|-5| = 5\)

The absolute value only considers magnitude, not direction. This property can sometimes be a bit confusing because people often relate numbers with their direction on the number line. In absolute value terms, we strip off the direction. This is the core property that helps in simplifying and solving absolute value equations.

Remember, no matter if you are starting with a positive or a negative number inside the absolute value bracket, its outcome is always directed towards being positive.

Equation Solving Steps

To solve an equation involving absolute value, there are specific steps we follow. Let's illustrate these with the equation \(-|2x + 1| = -3\).

• Recognize the Absolute Value Expression: The equation starts with an absolute value, and in our example, understanding that \(-|2x+1| = -3\) breaks down to \(|2x+1| = 3\) by removing the negative sign. This helps us set up equations that we can solve further.
• Set Up Two Equations: Since an absolute value can equal both a positive number and its negative counterpart, we set \(2x+1 = 3\) and \(2x+1 = -3\). These are the potential scenarios for \(|2x+1|\) equating to 3. This step reflects how absolute values measure distance.
• Solve Each Equation: Solve \(2x+1 = 3\) to find \(x = 1\) by isolating \(x\) through subtraction and division.
  Solve \(2x+1 = -3\) to find \(x = -2\) by performing similar operations as the first equation. These steps isolate the variable, giving us solutions to check later.

Following these steps ensures that all possible solutions are considered because of absolute value's nature.

Verification of Solutions

After solving the absolute value equation, it's vital to verify the solutions found. This step eliminates any potential errors that might have occurred during setup or calculation.

• For \(x = 1\), substituting back, \(-|2(1) + 1| = -3\), which simplifies to \(-|3| = -3\). This confirms our solution.


• For \(x = -2\), substitute back, \(-|2(-2) + 1| = -3\), which simplifies to \(-|-3| = -3\). This also validates the solution.

Verification is crucial since it confirms that the solutions satisfy the original equation. This ensures both solutions are accurate and adhere to the rule of absolute value being non-negative in its representation.