Problem 36 - College Algebra

Question

For the following exercises, solve the equation involving absolute value. $$ -|2 x+1|=-3 $$

Step-by-Step Solution

Answer

The solutions are \(x = 1\) and \(x = -2\).

Step 1: Understand the problem

We need to solve the equation involving an absolute value: \[-|2x+1|=-3\].This means we need to find the value of \(x\) that satisfies the equation.

Step 2: Eliminate negative sign from absolute value

Multiply both sides of the equation by -1 to remove the negative sign, resulting in:\[|2x+1| = 3\].This equation implies that the expression inside the absolute value can be either 3 or -3.

Step 3: Solve the first case

Solve the equation where the expression equals positive 3:\[2x+1 = 3\].Subtract 1 from both sides:\[2x = 2\].Divide by 2:\[x = 1\].This gives one solution, \(x = 1\).

Step 4: Solve the second case

Solve the equation where the expression equals negative 3:\[2x+1 = -3\].Subtract 1 from both sides:\[2x = -4\].Divide by 2:\[x = -2\].This gives the second solution, \(x = -2\).

Step 5: Verify the solutions

Substitute \(x = 1\) into the original equation to check:\[-|2(1)+1| = -|3| = -3\] which is true.Substitute \(x = -2\) into the original equation:\[-|2(-2)+1| = -|-3| = -3\] which is also true.Both \(x = 1\) and \(x = -2\) satisfy the original equation.

Key Concepts

Solving Absolute Value Equations

To solve equations involving absolute values, it's vital to understand what absolute value represents. The absolute value of a number reflects its distance from zero on a number line, always as a non-negative quantity. In particular, the equation \[|2x+1| = 3\] tells us that \(2x+1\) could be either 3 or -3. This is because the absolute value makes both \(3\) and \(-3\) equivalent in terms of distance from zero.

When solving absolute value equations:

• Identify the form \[|expression| = a\] and recognize that \[(expression) = a\] or \[(expression) = -a\].
• Split the equation into two separate cases to find all possible solutions.

Approaching the problem with this knowledge allows you to consider both potential scenarios that satisfy the absolute value equation.

Step-by-Step Algebra Solutions

Breaking down the solution into manageable steps can transform a complex equation into a series of achievable tasks. This clarity helps demystify solving processes, especially in algebra.

Here's how you can apply a step-by-step approach to the given exercise:

• Step 1: Recognize the need to manipulate the equation \(-|2x+1|=-3\) to remove negative signs. Multiply both sides by -1, resulting in \[|2x+1| = 3\].
• Step 2: Break it into cases: \[2x+1=3\] and \[2x+1=-3\]. Solve separately:
• For \[2x+1=3\], subtract \(1\) and divide by \(2\): \[x = 1\].
• For \[2x+1=-3\], subtract \(1\) and divide by \(2\): \[x = -2\].
• Step 3: Verification is crucial. Substitute your solutions back into the original equation to confirm they satisfy it.

Such ordered steps in solving an equation not only reinforce understanding but also build confidence in tackling diverse algebra problems.

College Algebra Problems

College algebra often presents problems that test your understanding of fundamental concepts like absolute values. These problems require you to intelligently apply your knowledge to find solutions.

When working with college-level algebra problems involving absolute values:

• Embrace the importance of verifying solutions. Mistakes can occur, and double-checking ensures your final answers are correct.
• Develop problem-solving strategies that rely on principles rather than memorization. Understanding each step in a solution strengthens your ability to tackle similar future problems.
• Practice regularly with a variety of problems, moving from simpler exercises to more complex ones as you build confidence.

Each problem solved deepens your comprehension and enhances your ability to work through advanced algebraic equations.