Problem 12 - Algebra 2 and Trigonometry

Question

In \(3-17,\) solve each equation or inequality. Each solution is an integer. $$ |2 x+3|-8=15 $$

Step-by-Step Solution

Answer

The solutions are \(x = 10\) and \(x = -13\).

Step 1: Isolate the Absolute Value

Add 8 to both sides of the equation to isolate the absolute value term. \[|2x + 3| - 8 + 8 = 15 + 8\]Simplifying gives:\[|2x + 3| = 23\]

Step 2: Remove the Absolute Value

To solve \(|2x + 3| = 23\), split it into two separate equations to remove the absolute value:1. \(2x + 3 = 23\)2. \(2x + 3 = -23\)

Step 3: Solve the First Equation

For the equation \(2x + 3 = 23\):Subtract 3 from both sides:\[2x = 23 - 3\]\[2x = 20\]Divide both sides by 2:\[x = \frac{20}{2} = 10\]

Step 4: Solve the Second Equation

For the equation \(2x + 3 = -23\):Subtract 3 from both sides:\[2x = -23 - 3\]\[2x = -26\]Divide both sides by 2:\[x = \frac{-26}{2} = -13\]

Step 5: Verify the Solutions

Substitute \(x = 10\) and \(x = -13\) back into the original equation to ensure they satisfy it.For \(x = 10\):\[|2(10) + 3| - 8 = 15 \Rightarrow |20 + 3| - 8 = 15 \Rightarrow 23 - 8 = 15 \Rightarrow 15 = 15\]For \(x = -13\):\[|2(-13) + 3| - 8 = 15 \Rightarrow |-26 + 3| - 8 = 15 \Rightarrow |-23| - 8 = 15 \Rightarrow 23 - 8 = 15 \Rightarrow 15 = 15\]

Key Concepts

Solving Equations

When you encounter an equation involving absolute values, solving it requires a straightforward approach. Here's how it works: Equations might sometimes seem complicated, but breaking them down into simple steps makes them a lot easier to handle. The main goal is to eliminate the absolute value signs, which allows us to work with regular equations.
A useful way to approach any equation is:

• Identify the absolute value expressions.
• Isolate them if necessary.
• Split the problem into two separate equations without absolute values.

By converting absolute value equations into two separate cases (positive and negative scenarios), we simplify what can look like a complex problem into something more manageable.

Integer Solutions

In these types of exercises, the solutions we are looking for are specifically integers. An integer is simply a whole number—this could be positive, negative, or zero.
For the given problem, after isolating and solving the equation, you correctly find that the solutions are integers: 10 and -13. Working with integer solutions is usually straightforward since they don't incorporate fractions or decimals.
When you resolve both split equations from an absolute value setup, it's a good practice to double-check if what you find adheres strictly to the condition of being whole numbers.

Isolating the Absolute Value

Isolating the absolute value is an essential step when solving such equations. It means rearranging the equation so that the absolute value expression is by itself on one side.
Here’s how it was done in our example: initially, the absolute value expression \(|2x + 3|\) was combined with other operations. By adding 8 to both sides, it results in:\[|2x + 3| = 23\]
This isolation simplifies the process ahead and allows us to confidently proceed to the next step, which involves removing the absolute value bars. Remember, always adjust both sides before solving further, keeping the equation balanced.

Verifying Solutions

Verifying your solutions ensures that the solutions satisfy the original equation. It’s like a double-check to confirm the work is right! Substituting back into the equation checks for errors.
Let’s revisit our example solutions of \(x = 10\) and \(x = -13\). By plugging them back into the position of \(x\) in the equation, we verify if they both lead to a true mathematical statement.

• For \(x = 10\), the left side should equal the right side 15.
• For \(x = -13\), the same must occur, validating both solutions effectively.

Ultimately, this step is about reinforcing the accuracy and integrity of your solutions. It reassures you that you’ve followed each step correctly and achieved valid results.