Problem 12 - Algebra for College Students

Question

Solve each equation. $$ |2 x+3|=19 $$

Step-by-Step Solution

Answer

x = 8 and x = -11

Step 1: Understand the Absolute Value

The equation involves an absolute value, which means that \(2x + 3\) could be either positive or negative but still result in 19. So, set up two separate equations to consider both possibilities.

Step 2: Set Up Two Equations

Write two equations: one where \(2x + 3 = 19\) and another where \(2x + 3 = -19\). This accounts for both the positive and negative scenarios.

Step 3: Solve the Positive Scenario

For the first equation, \(2x + 3 = 19\). Subtract 3 from both sides to get \(2x = 16\). Then, divide both sides by 2 to get \(x = 8\).

Step 4: Solve the Negative Scenario

For the second equation, \(2x + 3 = -19\). Subtract 3 from both sides to get \(2x = -22\). Then, divide both sides by 2 to get \(x = -11\).

Step 5: Verify the Solutions

Check if \(x = 8\) and \(x = -11\) are indeed solutions by substituting them back into the original equation to ensure both satisfy \(|2x + 3| = 19\). Both values work.

Key Concepts

Solving Equations

Solving equations is one of the most fundamental skills in algebra. An equation states that two expressions are equal, and solving it means finding the value of the variable that makes this statement true. When dealing with equations, you often have to perform operations like addition, subtraction, multiplication, and division on both sides of the equation. These operations help us isolate the variable, thereby finding its value. In the case of absolute value equations, additional steps are necessary to handle the absolute value property.

Absolute Value Properties

The absolute value of a number is its distance from zero on the number line, regardless of direction. It is always non-negative. For any real number, the absolute value is denoted by two vertical bars, like \( |a| \). Some key properties of absolute value are:

• \( |a| \geq 0 \) : Absolute value is always non-negative.
• \( |a| = a \) if \( a \geq 0 \)
• \( |a| = -a \) if \( a < 0 \)
• For any absolute value equation \( |A| = B \), it can be rewritten into two separate equations: A = B and A = -B.

Understanding these properties is crucial for solving absolute value equations. It allows us to consider both the positive and negative scenarios that could satisfy the equation.

Step-by-Step Solutions

Let's apply these concepts to solve the given absolute value equation \( |2x + 3| = 19 \). Follow these steps:
1. **Understand the Absolute Value:** Recognize that \( 2x + 3 \) could be either 19 or -19.
2. **Set Up Two Equations:** Write two equations: \( 2x + 3 = 19 \) and \( 2x + 3 = -19 \). This accounts for both possibilities of the absolute value.
3. **Solve the Positive Scenario:** For \( 2x + 3 = 19 \):

• Subtract 3 from both sides: \( 2x = 16 \)
• Divide both sides by 2: \( x = 8 \)

4. **Solve the Negative Scenario:** For \( 2x + 3 = -19 \):

• Subtract 3 from both sides: \( 2x = -22 \)
• Divide both sides by 2: \( x = -11 \)

5. **Verify the Solutions:** Plug \( x = 8 \) and \( x = -11 \) back into the original equation \( |2x + 3| = 19 \) to ensure both values satisfy it. Indeed, both values work perfectly and hold true for the absolute value equation.
Understanding the step-by-step process for solving absolute value equations makes it easier to tackle similar problems.