Problem 53 - A Graphical Approach to Precalculus with Limits

Question

Solve each group of equations and inequalities analytically. (a) \(|7-2 x|=3\) (b) \(|7-2 x| \geq 3\) (c) \(|7-2 x| \leq 3\)

Step-by-Step Solution

Answer

(a) \(x = 2, 5\); (b) \(x \leq 2\) or \(x \geq 5\); (c) \(2 \leq x \leq 5\).

Step 1: Understand the Absolute Value Equation

For the equation \(|7-2x| = 3\), we need to consider two cases because the absolute value of an expression equals a number when the expression itself is either equal to the number or its negative. Therefore, the equation splits into two cases: 1. \(7 - 2x = 3\)2. \(7 - 2x = -3\).

Step 2: Solve Case 1 of Equation (a)

Solve the equation \(7 - 2x = 3\).- Subtract 7 from both sides: \(-2x = 3 - 7\)- Simplify: \(-2x = -4\)- Divide both sides by -2: \(x = 2\).

Step 3: Solve Case 2 of Equation (a)

Solve the equation \(7 - 2x = -3\).- Subtract 7 from both sides: \(-2x = -3 - 7\)- Simplify: \(-2x = -10\)- Divide both sides by -2: \(x = 5\).

Step 4: Interpret Solutions for Equation (a)

The solutions for the equation \(|7-2x|=3\) are \(x = 2\) and \(x = 5\).

Step 5: Understand the Absolute Value Inequality (b)

For \(|7-2x| \geq 3\), we have two cases to consider: 1. \(7 - 2x \geq 3\)2. \(7 - 2x \leq -3\).

Step 6: Solve Inequality Case 1 of (b)

Solve the inequality \(7 - 2x \geq 3\).- Subtract 7 from both sides: \(-2x \geq 3 - 7\)- Simplify: \(-2x \geq -4\)- Divide both sides by -2 and reverse the inequality: \(x \leq 2\).

Step 7: Solve Inequality Case 2 of (b)

Solve the inequality \(7 - 2x \leq -3\).- Subtract 7 from both sides: \(-2x \leq -3 - 7\)- Simplify: \(-2x \leq -10\)- Divide both sides by -2 and reverse the inequality: \(x \geq 5\).

Step 8: Interpret Solutions for Inequality (b)

The solutions for the inequality \(|7-2x| \geq 3\) are \(x \leq 2\) or \(x \geq 5\).

Step 9: Understand the Absolute Value Inequality (c)

For \(|7-2x| \leq 3\), similar to before, consider the cases: 1. \(7 - 2x \leq 3\)2. \(7 - 2x \geq -3\).

Step 10: Solve Inequality Case 1 of (c)

Solve the inequality \(7 - 2x \leq 3\).- Subtract 7 from both sides: \(-2x \leq 3 - 7\)- Simplify: \(-2x \leq -4\)- Divide both sides by -2 and reverse the inequality: \(x \geq 2\).

Step 11: Solve Inequality Case 2 of (c)

Solve the inequality \(7 - 2x \geq -3\).- Subtract 7 from both sides: \(-2x \geq -3 - 7\)- Simplify: \(-2x \geq -10\)- Divide both sides by -2 and reverse the inequality: \(x \leq 5\).

Step 12: Interpret Solutions for Inequality (c)

The solutions for the inequality \(|7-2x| \leq 3\) are for \(2 \leq x \leq 5\).

Key Concepts

Solving Equations

Equations involving absolute values might seem tricky at first, but they can be tackled by breaking them down into simpler cases. Consider the absolute value equation \(|7-2x| = 3\).This tells us that the expression inside the absolute value, \(7 - 2x\), can be either 3 or -3.

• Case 1: Solve for when \(7 - 2x = 3\). Subtracting 7 from both sides gives us \(-2x = -4\). Dividing by -2, we find that \(x = 2\).
• Case 2: Solve for when \(7 - 2x = -3\). Similarly, subtract 7 to get \(-2x = -10\). Again, dividing by -2, we get \(x = 5\).

Hence, the solutions to the equation \(|7-2x| = 3\) are \(x = 2\) and \(x = 5\). Solving equations in this way ensures comprehensive understanding by considering both potential scenarios within the absolute value expression.

Inequalities

Solving absolute value inequalities requires recognizing that the expression can be greater or lesser than a specified number. Let's first examine the inequality \(|7-2x| \geq 3\).For absolute value inequalities:

• Case 1: Assume \(7 - 2x \geq 3\). Subtracting 7 on both sides results in \(-2x \geq -4\). After dividing by -2, remembering to flip the inequality sign, we obtain \(x \leq 2\).
• Case 2: Assume \(7 - 2x \leq -3\). Similarly, subtracting 7 yields \(-2x \leq -10\). Dividing by -2 and reversing the inequality provides \(x \geq 5\).

Therefore, the inequality \(|7-2x| \geq 3\) can be expressed in terms of \(x \leq 2\) or \(x \geq 5\).For the inequality \(|7-2x| \leq 3\), apply a similar approach:

• Case 1: Consider \(7-2x \leq 3\). Subtract 7 to get \(-2x \leq -4\). Dividing by -2 (and flipping the sign) results in \(x \geq 2\).
• Case 2: Consider \(7-2x \geq -3\). Subtract 7 to find \(-2x \geq -10\), and dividing by -2 gives \(x \leq 5\).

This results in the solution \(2 \leq x \leq 5\) for \(|7-2x| \leq 3\), which means \(x\) values are within this closed interval.

Analytical Methods

Analytical methods allow us to solve equations and inequalities with precision by systematically following a set of logical steps. This method is particularly useful for absolute value problems, which are based on expressions that represent distance from zero on a number line. Starting with an equation or inequality, we always ensure to:

• Break down the problem into manageable cases: Recognize that the function inside the absolute value can be positive or negative, leading to two separate linear equations or inequalities to solve.
• Solve each case independently: Treat each scenario as its own problem. This often involves performing operations such as addition, subtraction, and division, while being mindful to reverse the inequality sign when dividing by a negative number.
• Combine solutions logically: Once each case is solved, the results are interpreted either as separate solutions (for equations) or intervals that may overlap or be combined in a logical fashion (for inequalities).

Using analytical methods means engaging closely with the mathematical properties of absolute values and ensures that all potential solutions are explored and interpreted correctly. This approach not only finds solutions but bolsters understanding of the mathematical reasoning behind each step.