Problem 36 - Algebra 2

Question

Solve each equation. Check your solutions. \(-9=-3|2 n+5|\)

Step-by-Step Solution

Answer

The solutions are \(n = -1\) and \(n = -4\).

Step 1: Understand the Absolute Value Equation

Given the equation \(-9 = -3|2n + 5|\). The absolute value equation is of the form \(-a = -b|c|\)."

Step 2: Eliminate Negative Coefficient

Divide both sides by \(-3\) to get \(3 = |2n + 5|\). This isolates the absolute value.

Step 3: Set Up Two Cases

Since \(|x| = a\) implies \(x = a\) or \(x = -a\), set up two equations: \(2n + 5 = 3\) and \(2n + 5 = -3\).

Step 4: Solve Case 1

Solve the equation \(2n + 5 = 3\). Subtract \(5\) from both sides to obtain \(2n = -2\). Next, divide by \(2\) to get \(n = -1\).

Step 5: Solve Case 2

Solve the equation \(2n + 5 = -3\). Subtract \(5\) from both sides to obtain \(2n = -8\). Divide by \(2\) to get \(n = -4\).

Step 6: Check Solutions

Substitute \(n = -1\) back into the original equation: \(-9 = -3|2(-1) + 5| \rightarrow -9 = -3|3| \rightarrow -9 = -9\). This is correct. Next, substitute \(n = -4\): \(-9 = -3|2(-4) + 5| \rightarrow -9 = -3|-3| \rightarrow -9 = -9\). This is also correct.

Key Concepts

Solving Equations

To solve an absolute value equation like \(-9 = -3|2n + 5|\), we first need to isolate the absolute value expression. The goal here is to simplify any coefficient outside the absolute value bars. This simplifies the equation greatly.

Start by dividing both sides by \(-3\). This action is crucial because it transforms the equation into a more manageable form: \(3 = |2n + 5|\). Once the absolute value is isolated, the next step involves creating two separate equations to solve. This is because \(|x| = a\) implies two potential solutions, \(x = a\) and \(x = -a\), due to the nature of absolute values.

• Set up \(2n + 5 = 3\)
• Set up \(2n + 5 = -3\)

Solving these two simpler equations will yield the potential values for \(n\). Each case reflects a different possible scenario given by the absolute value property.

Checking Solutions

Whenever you solve an equation, especially involving absolute values, it's important to check your solutions by plugging them back into the original equation. This confirms correctness and checks any possible errors made during the solving steps.

To verify the solution for \(n = -1\), substitute it back: \(-9 = -3|2(-1) + 5|\). Substitute and simplify inside the absolute value to get \(-9 = -3|3|\), which holds true as \(-9 = -9\).

• Next, check \(n = -4\). Substitute: \(-9 = -3|2(-4) + 5|\). Simplifying gives \(-9 = -3|-3|\). This also holds as \(-9 = -9\).

Checking ensures that both solutions are valid, showing that no error occurred in each calculation step. This is a standard practice to maintain accuracy.

Algebraic Manipulation

Algebraic manipulation is the process of using various algebraic methods to simplify or solve an equation. In this context, it involves rearranging and simplifying the equation in several steps.

When dealing with absolute values like \(-9 = -3|2n + 5|\), algebraic manipulation begins by first isolating the absolute value expression through division. This turns \(-9 = -3|2n + 5|\) into \(3 = |2n + 5|\).

• Next, solve each equation derived from removing the absolute value by isolating \(n\). For \(2n + 5 = 3\), subtract \(5\) and divide by \(2\) gives \(n = -1\).
• For \(2n + 5 = -3\), subtract \(5\) and divide by \(2\) yields \(n = -4\).

These steps illustrate the power of algebraic manipulation in breaking down complex-looking equations into simpler, manageable parts that anyone can solve with practice.