Problem 10 - Algebra 2 and Trigonometry

Question

In \(3-17,\) solve each equation or inequality. Each solution is an integer. $$ |3-y|=8 $$

Step-by-Step Solution

Answer

The integer solutions are \(y = -5\) and \(y = 11\).

Step 1: Express the Absolute Value Equation as Two Separate Equations

The equation \(|3-y|=8\) can be rewritten as two separate equations since the expression inside the absolute value can be either positive or negative. Thus, we have:1. \(3-y=8\)2. \(3-y=-8\).

Step 2: Solve the First Equation

Solve the equation \(3-y=8\).- Subtract 3 from both sides: \(3 - y - 3 = 8 - 3\)- Simplify to find \(-y = 5\).- Multiply both sides by -1 to solve for \(y\): \(y = -5\).

Step 3: Solve the Second Equation

Solve the equation \(3-y=-8\).- Subtract 3 from both sides: \(3 - y - 3 = -8 - 3\)- Simplify to find \(-y = -11\).- Multiply both sides by -1 to solve for \(y\): \(y = 11\).

Step 4: Validate the Solutions

Substitute the solutions back into the original equation to check if they are correct:1. Substitute \(y = -5\) into \(|3 - y| = 8\): \(|3 - (-5)| = |3 + 5| = 8\) which is true.2. Substitute \(y = 11\) into \(|3 - y| = 8\): \(|3 - 11| = |-8| = 8\) which is also true.Both solutions satisfy the original equation.

Key Concepts

Integer Solutions

In the realm of mathematics, integer solutions are particularly significant because they are whole numbers that can be either positive, negative, or zero. These types of solutions are crucial when dealing with equations that apply to real-world scenarios, such as counting objects or discrete elements.

For absolute value equations like \(|3-y| = 8\), the problem asks for integer solutions, meaning we need to find values of \(y\) that satisfy the equation where \(y\) is an integer.

It is important to remember that the absolute value of any number is the distance it is from zero on the number line, so it is always non-negative. As a result, when solving absolute value equations, we usually end up with two potential solutions, reflecting the positive and negative scenarios of the expression inside the absolute value.

Solving Equations

Solving equations is a fundamental aspect of mathematics, and understanding how to handle absolute value equations is key. Let's dive into how to solve an equation like \(|3-y| = 8\).

**Step 1: Interpret the Absolute Value**
First, realize that absolute value equations split into two cases because the quantity inside the absolute value \(3-y\) could either be 8 or -8. Therefore, we rewrite it as two separate equations: \(3-y = 8\) and \(3-y = -8\).

**Step 2: Solve Each Equation**
Solve each equation separately:

• Equation 1: \(3 - y = 8\). To isolate \(y\), subtract 3 from both sides to get \(-y = 5\). Then multiply by -1 to find \(y = -5\).
• Equation 2: \(3-y=-8\). Similarly, subtract 3 from both sides to achieve \(-y = -11\), and then multiply by -1 to conclude \(y = 11\).

Remember that solving each separate equation gives us the complete set of solutions that satisfy the original absolute value equation.

Validating Solutions

Once we have potential solutions, validating them ensures our answers meet the original problem's requirements. In this case, after determining \(y = -5\) and \(y = 11\) as solutions for the equation \(|3-y| = 8\), it's important to check them by substitution.

**Step 1: Substitution into the Original Equation**

• Substitute \(y = -5\) into the equation: \(|3 - (-5)| = |3 + 5| = 8\). This validates the solution since it simplifies to \(8 = 8\).
• Substitute \(y = 11\) into the equation: \(|3 - 11| = |-8| = 8\), which also checks out because it simplifies to \(8 = 8\).

Verifying solutions by plugging them back into the original equation ensures that both solutions are correct and that no errors occurred during solving. Hence, both \(y = -5\) and \(y = 11\) are indeed the valid integer solutions to the equation.