Problem 60 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation or inequality. $$|2 x+4|+2=10$$

Step-by-Step Solution

Answer

The solutions are \(x = 2\) and \(x = -6\).

Step 1: Isolate the Absolute Value Expression

Start by isolating the absolute value expression \(|2x + 4|\). To do this, subtract 2 from both sides of the equation \(|2x + 4| + 2 = 10\):\[ |2x + 4| = 10 - 2 \]which simplifies to:\[ |2x + 4| = 8 \]

Step 2: Remove the Absolute Value

Set up two separate equations to solve for \(x\) since \(|A| = B\) implies \(A = B\) or \(A = -B\). Thus:1. \(2x + 4 = 8\)2. \(2x + 4 = -8\)

Step 3: Solve the First Equation

Solve the equation \(2x + 4 = 8\):Subtract 4 from both sides:\[ 2x = 8 - 4 \]\[ 2x = 4 \]Divide by 2:\[ x = \frac{4}{2} \]\[ x = 2 \]

Step 4: Solve the Second Equation

Solve the equation \(2x + 4 = -8\):Subtract 4 from both sides:\[ 2x = -8 - 4 \]\[ 2x = -12 \]Divide by 2:\[ x = \frac{-12}{2} \]\[ x = -6 \]

Step 5: Verify the Solutions

Verify both solutions by substituting them back into the original equation to ensure they satisfy it.- For \(x = 2\): \[ |2(2) + 4| + 2 = |4 + 4| + 2 = |8| + 2 = 8 + 2 = 10 \] The solution \(x = 2\) is correct.- For \(x = -6\): \[ |2(-6) + 4| + 2 = |-12 + 4| + 2 = |-8| + 2 = 8 + 2 = 10 \] The solution \(x = -6\) is correct.

Key Concepts

Equation Solving

Solving equations is all about finding the value of the variable that makes the equation true. In our given exercise, we are dealing with an absolute value equation. An absolute value equation involves an expression within absolute value bars, which is always non-negative. The goal is to solve it so that the variable, in this case, \(x\), results in a true statement when substituted back into the original equation.When you see an absolute value equation like \(|2x + 4| + 2 = 10\), your first step should be to focus on the expression inside the absolute value and think about the two possible scenarios: - The expression inside can equal the number on the other side (e.g., \(8\)), or - It can equal the negative of that number (e.g., \(-8\)).This duality is because the absolute value "hides" the difference between positive and negative values. You'll set up two separate equations to solve.

Isolating Variables

The process of isolating variables is like unwrapping a present, revealing what the variable is hiding inside. Our variable of interest here is \(x\). The exercise starts with a slight twist since we have to first isolate the absolute value expression before tackling \(x\) directly.Isolating \(|2x + 4|\) is achieved by reversing additional operations around it. This means removing or "undoing" the plus 2 on the left side. We do this by subtracting 2 from both sides of the original equation \(|2x + 4| + 2 = 10\), which simplifies the equation to \(|2x + 4| = 8\). Now, the absolute value is all by itself on one side!At this stage, you're ready to address the specific values hidden within the absolute value by solving two separate equations, thereby solving for \(x\) directly.- \(2x + 4 = 8\) leads to an easy solution path for \(x = 2\)- \(2x + 4 = -8\) directs you to find \(x = -6\)These steps involve simply undoing the operations in the order opposite to how they would naturally apply: add/subtract first, and then multiply/divide if needed.

Verification of Solutions

The final, crucial step in solving equations is verifying your solutions, ensuring that they actually satisfy the original equation. This ensures no mistakes were made along the way. Verification is simple yet powerful.Once you have computed solutions \(x = 2\) and \(x = -6\), substituting them back into the original equation should retrace the steps back to a true statement. Let’s verify:1. For \(x = 2\), plug this back into \(|2(2) + 4| + 2 = 10\). Calculating the expression inside, \(|4 + 4| = |8|\), we end up with \(8 + 2 = 10\). It checks out!2. For \(x = -6\), substitute this into the original equation, you obtain \(|2(-6) + 4| + 2 = 10\). Calculating the expression inside, \(|-12 + 4| = |-8|\), results again in \(8 + 2 = 10\). This confirms the correctness.Both solutions satisfy the equation, ensuring they're valid. Verification not only assures correctness but deepens understanding and reinforces the step-by-step solution process.