Problem 24 - Algebra 2

Question

Solve each equation. Check your solutions. \(|y+9|=21\)

Step-by-Step Solution

Answer

The solutions are \(y = 12\) and \(y = -30\).

Step 1: Understand the Definition of Absolute Value

Absolute value represents the distance from zero on a number line, so \(|y+9| = 21\) implies that the expression inside the absolute value, \(y + 9\), can be either 21 or -21.

Step 2: Solve for Positive Case

Set up the equation for the positive case: \(y + 9 = 21\). Subtract 9 from both sides to isolate \(y\). Solve: \(y + 9 - 9 = 21 - 9\), which simplifies to \(y = 12\).

Step 3: Solve for Negative Case

Set up the equation for the negative case: \(y + 9 = -21\). Subtract 9 from both sides again. Solve: \(y + 9 - 9 = -21 - 9\), which simplifies to \(y = -30\).

Step 4: Check Solutions

Insert each solution back into the original equation to ensure they are correct.For \(y = 12\):\(|12 + 9| = |21| = 21\), which matches the original equation.For \(y = -30\):\(|-30 + 9| = |-21| = 21\), which also matches the original equation.

Key Concepts

Solving Absolute Value Equations

Solving equations involving absolute values requires understanding the notion of distance. The absolute value of a number is its distance from zero, which is never negative. This means equations of the form \(|x| = a\), where \(a\) is positive, lead to two possible scenarios.

To solve \(|y+9|=21\), you first consider the positive case, where the expression inside the absolute value is equal to 21. This gives us one equation:

• \(y + 9 = 21\) - Solving this straightforward equation involves subtracting 9 from both sides: \(y = 12\).

Likewise, consider the negative case, where the expression equals -21:

• \(y + 9 = -21\) - Here, you also subtract 9 from both sides to find \(y = -30\).

Considering Positive and Negative Cases

The inclusion of both positive and negative cases is fundamental to solving absolute value equations. Since absolute value measures distance only, both a positive and negative solution may exist relative to the zero point.

This dual nature explains why we derive two separate equations: one for when \(y + 9 = 21\), and another for \(y + 9 = -21\). Each scenario stems directly from the fundamental property of absolute value.
Here's why this approach is used:

• **Symmetry Property:** Both positive and negative values at the same distance from zero lead to the same absolute value.
• **Real-World Analogy:** If someone describes a distance of 21 units from a starting line, they could be 21 units forward or backward.

Considering both cases ensures you capture all possible solutions of \(|y + 9|=21\).

Checking Equation Solutions

After solving absolute value equations, checking your solutions is crucial to assure correctness. This involves substituting each solution back into the original equation to see if both sides are equal.

For \(y = 12\), substitute into the original equation:

• \(|12 + 9| = |21| = 21\) confirms the solution is valid.

Checking for the negative scenario with \(y = -30\):

• \(|-30 + 9| = |-21| = 21\) also verifies that \(y = -30\) is a valid solution.

This step prevents errors that can arise from algebraic manipulation and ensures that all solutions are correct with respect to the original equation.