Problem 49 - College Algebra with Modeling and Visualization

Question

Solve the equation symbolically. Then solve the related inequality. $$|2.1 x-0.7|=2.4, \quad|2.1 x-0.7| \geq 2.4$$

Step-by-Step Solution

Answer

The solution to the equation is \(x \approx 1.476\) or \(x \approx -0.810\). For the inequality, \(x \leq -0.810\) or \(x \geq 1.476\).

Step 1: Understand the Absolute Value Equation

The problem involves solving the absolute value equation \(|2.1x - 0.7| = 2.4\). An absolute value equation of the form \(|A| = B\) can be split into two separate equations: \(A = B\) and \(A = -B\). This means we'll have two equations to solve: \(2.1x - 0.7 = 2.4\) and \(2.1x - 0.7 = -2.4\).

Step 2: Solve the First Equation

Let's solve the first equation: \(2.1x - 0.7 = 2.4\).Add \(0.7\) to both sides to isolate the term with \(x\):\[ 2.1x = 3.1 \]Next, divide both sides by \(2.1\) to solve for \(x\):\[ x = \frac{3.1}{2.1} \approx 1.476 \]

Step 3: Solve the Second Equation

Now solve the second equation: \(2.1x - 0.7 = -2.4\).Add \(0.7\) to both sides:\[ 2.1x = -1.7 \]Then, divide both sides by \(2.1\) to find \(x\):\[ x = \frac{-1.7}{2.1} \approx -0.810 \]

Step 4: Solve the Absolute Value Inequality

The inequality \(|2.1x - 0.7| \geq 2.4\) implies two inequalities that we need to consider: \(2.1x - 0.7 \geq 2.4\) and \(2.1x - 0.7 \leq -2.4\).

Step 5: Solve the First Inequality

Solve the inequality \(2.1x - 0.7 \geq 2.4\):Add \(0.7\) to both sides:\[ 2.1x \geq 3.1 \]Divide each side by \(2.1\) to solve for \(x\):\[ x \geq \frac{3.1}{2.1} \approx 1.476 \]

Step 6: Solve the Second Inequality

Solve the inequality \(2.1x - 0.7 \leq -2.4\):Add \(0.7\) to both sides:\[ 2.1x \leq -1.7 \]Divide each side by \(2.1\) to solve for \(x\):\[ x \leq \frac{-1.7}{2.1} \approx -0.810 \]

Step 7: Write the Solution for the Inequality

The solution for the inequality is the combination of both conditions from the inequalities: \(x \leq -0.810\) or \(x \geq 1.476\). This describes the range where the inequality holds true.

Key Concepts

Inequality Solving

Inequality solving involves finding the range of values that satisfy the inequality condition.
This can often be more complex than simply finding solutions to equations. In this exercise, we have the inequality \(|2.1x - 0.7| \geq 2.4\).
The absolute value inequality means we have two situations to consider:

• \(2.1x - 0.7 \geq 2.4\)
• \(2.1x - 0.7 \leq -2.4\)

For these inequalities, we solve by isolating \(x\) just like with equations, but ensuring that we capture a range of solutions.
First, solve \(2.1x - 0.7 \geq 2.4\), which gives \(x \geq \frac{3.1}{2.1} \approx 1.476\).
Similarly, solve \(2.1x - 0.7 \leq -2.4\), resulting in \(x \leq \frac{-1.7}{2.1} \approx -0.810\).
This tells us that \(x\) is in the range \(x \leq -0.810\) or \(x \geq 1.476\). Inequality solutions can be seen on a number line to better understand the intervals where the values of \(x\) satisfy the given inequality.

Symbolic Computation

Symbolic computation means solving equations using symbols and algebraic manipulation rather than numerical approximations.
It plays a crucial role in solving the absolute value equation and inequality in this exercise. For the equation \(|2.1x - 0.7| =2.4\), symbolic computation involves breaking down the absolute value condition into two separate linear equations:

• \(2.1x - 0.7 = 2.4\)
• \(2.1x - 0.7 = -2.4\)

Each equation is solved symbolically, by using addition and division to isolate the variable \(x\).
Symbolic computation ensures that the structure and properties of the mathematical expressions are preserved, leading to accurate solutions.
It is especially beneficial when dealing with equations, as it allows for an understanding of how different components interact algebraically. The solutions derived using symbolic computation provide exact answers rather than approximations, which is crucial for understanding the behavior of more complex systems in mathematics.

Algebraic Expressions

Understanding algebraic expressions is essential for solving both equations and inequalities.
In this problem, the expression \(2.1x - 0.7\) within the absolute value plays a significant role. An algebraic expression includes variables, constants, and operations like addition and multiplication.
In this case, the expression describes a line when graphed.
Handling the expression \(2.1x - 0.7\) requires knowing how to manipulate terms. When solving for \(x\), we add or subtract constants and multiply or divide by other constants to simplify the expression.
This understanding allows us to perform operations to isolate \(x\).
Such manipulation is key to solving both parts of the problem, the equation \(|2.1x - 0.7| = 2.4\) and the inequality \(|2.1x - 0.7| \geq 2.4\), where isolating the algebraic expression is necessary to find the correct solution values.
Simply put, mastering algebraic expressions builds confidence in solving a wide range of mathematical problems smoothly!