Problem 62 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation or inequality. $$5|x+3|-2=18$$

Step-by-Step Solution

Answer

The solutions are \(x = 1\) and \(x = -7\).

Step 1: Isolate the Absolute Value

Begin by adding 2 to both sides of the equation to isolate the absolute value.\[ 5|x+3| - 2 = 18 \5|x+3| = 20 \]

Step 2: Divide to Simplify

Now, divide both sides by 5 to solve for the absolute value expression:\[ |x+3| = 4 \]

Step 3: Consider the Positive Case

Set up the first equation assuming the expression inside the absolute value is positive:\[ x + 3 = 4 \x = 4 - 3 \x = 1 \]

Step 4: Consider the Negative Case

Set up the second equation assuming the expression inside the absolute value is negative:\[ x + 3 = -4 \x = -4 - 3 \x = -7 \]

Step 5: Solution Summary

Thus, the two solutions for this equation are:\[ x = 1 \text{ or } x = -7 \]

Key Concepts

Solving Equations

Solving equations involves finding the values of unknown variables that make the equation true. When solving equations that include absolute values, our goal is to isolate the absolute value expression on one side. Absolute value equations represent the distance from zero, which means they present two potential results: one positive and one negative. Begin solving absolute value equations by following these general steps:

• Isolate the absolute value on one side.
• Consider both the positive and negative scenarios.
• Solve each resulting equation for the unknown variable.

These steps ensure that all possible solutions are found. Understanding these principles is crucial when tackling physics, engineering, or any field involving real-world calculations.

Algebraic Expressions

Algebraic expressions are foundational in solving equations. In the given problem, the expression inside the absolute value is \(x + 3\). An algebraic expression can be a combination of numbers, variables, and operations (e.g., addition, subtraction). Understanding the components:

• **Constants**: fixed numbers in an expression. Here, the numbers are 3, 5, and 2.
• **Variables**: symbols representing unknown values. In this case, the variable is \(x\).
• **Operations**: mathematical processes (addition, subtraction, etc.).

The ability to manipulate algebraic expressions through operations like addition and division is key. This allows for isolating variables and simplifying expressions, creating new expressions that are easier to solve.

Inequalities

Though the given exercise is an equation, understanding inequalities is essential as they often accompany absolute value equations. Inequalities involve expressions that use symbols like \( >, <, \geq, \leq \). They define a range of possible solutions rather than specific values:

• Solving inequalities involves similar steps to equations: isolate the variables, then consider all possible solutions.
• Remember that flipping the inequality sign is necessary when multiplying or dividing by a negative number.
• Inequalities are used to express solutions of absolute value expressions when the results aren't perfect numbers.

Cognitive grasp on inequalities contributes to enhanced proficiency in handling real-life problem-solving scenarios.

Step-by-Step Solutions

Step-by-step solutions play a pivotal role in understanding complex mathematical problems like absolute value equations. By breaking down each stage of the solution:

• **Step 1**: Identifies the necessity to isolate the absolute value.
• **Step 2**: Involves simplifying by dividing, making the problem more approachable.
• **Steps 3 and 4**: Discuss both the positive and negative scenarios, covering all potential outcomes.
• **Step 5**: Summarizes the solutions obtained, reaffirming understanding.

Each step serves as a building block leading to the final solution. This method reassures learners that no detail is missed, promoting confidence. Follow through each part of the solution meticulously, replicating the approach to other questions.