Problem 69 - Algebra for College Students

Question

Solve each equation. \(|-2 x-3|=|x+1|\)

Step-by-Step Solution

Answer

Solutions: \(x = -\frac{4}{3}\) and \(x = -2\).

Step 1: Understand the Absolute Value Equation

The equation is \[|-2x - 3| = |x + 1|\].Absolute value equations such as this one require considering both the positive and negative cases for each side of the equation. We will solve it by considering different scenarios for the expressions inside the absolute values.

Step 2: Consider the Positive Case

In this case, both expressions are positive, which gives the equation:\[-2x - 3 = x + 1\]. To solve for x, add 2x to both sides:\[-3 = 3x + 1\]. Subtract 1 from both sides:\[-4 = 3x\]. Finally, divide each side by 3 to isolate x:\[x = -\frac{4}{3}\].

Step 3: Consider the Negative Case of the Left Expression

In this scenario, the left-hand side is the negative of the right, giving:\[-(-2x - 3) = x + 1\], which simplifies to:\[2x + 3 = x + 1\]. Subtract x from both sides:\[x + 3 = 1\]. Then subtract 3 from both sides:\[x = -2\].

Step 4: Consider the Negative Case of the Right Expression

Here, the right expression is negative, leading to:\[-2x - 3 = -(x + 1)\], which simplifies to:\[-2x - 3 = -x - 1\]. Add x to both sides:\[-x - 3 = -1\]. Then add 3 to both sides:\[-x = 2\]. Multiply both sides by -1 to solve for x:\[x = -2\]. Note that this yields the same solution as the previous scenario, which is consistent due to overlapping regions of inequality.

Step 5: Verify Solutions and Write Conclusion

We found two potential solutions: \[x = -\frac{4}{3}\] and \[x = -2\]. Substitute these back into the original equation to verify. For \[x = -\frac{4}{3}\], both sides evaluate to \[\frac{5}{3}\]. For \[x = -2\], both sides evaluate to \[1\]. Thus, both values satisfy the original equation, confirming our solutions.

Key Concepts

Solving Equations

Equations are mathematical statements that show that two expressions are equal. Solving equations is about finding the value of the variable that makes this statement true. In the case of absolute value equations, like \(|-2x - 3| = |x + 1|\), solving involves considering different scenarios due to the nature of absolute values. Absolute value measures the distance of a number from zero on the number line, regardless of direction. Hence, both positive and negative cases must be evaluated.

• For the positive case, assume both expressions equal their positive forms.
• Switching to the negative scenarios helps disclose all potential solutions.

This logical approach ensures that all possibilities are taken into account, thereby providing a complete set of solutions to the equation.

Algebra Concepts

Algebra is more than just manipulating symbols. It involves understanding how numbers and variables relate within an equation. In the given exercise, recognizing that the symbols |...| represent absolute values is crucial. It leads to forming different cases, which ultimately helps in solving the equation.

• Understand the role of equal expressions formed by the absolute values.
• Break the equation into simpler parts to solve step by step.
• Utilize inverse operations like addition, subtraction, multiplication, and division.

Algebra brings a toolset that simplifies complex expressions and provides clarity on how to approach each part of the equation logically. It's this framework that guides us through multi-step problems with confidence.

College Algebra

College algebra pulls together different mathematical concepts that are well beyond basic high school algebra. It requires a deeper understanding of relations and functions. When tasked with solving an equation involving absolute values like our example, students in college algebra must recognize that it extends beyond routine arithmetic and algebraic procedures.

• Develop skills for abstract thinking and equation manipulation.
• Ability to dissect an equation into manageable pieces.
• Confidence in evaluating different scenarios to find all possible solutions.

This level of algebra is pivotal in establishing a solid foundation for more advanced studies like calculus, bridging the gap with comprehensive equation-solving strategies.

Equation Verification

Solving an equation does not end when you find potential solutions. Equation verification is essential to ensure your solutions are correct. It involves substituting the solutions back into the original equation to check their validity.

• Insert each solution individually into the original equation.
• Ensure that both sides of the equation equate upon substitution.
• Dismiss any solutions that do not fulfill the original equation.

In our exercise, solutions \(x = -\frac{4}{3}\) and \(x = -2\) were checked by substituting them into the initial equation. Both substitutions resulted in equal values on both sides, verifying their accuracy. This step is vital for confidence in your solution, reassuring you that the values derived actually satisfy the equation.