Problem 41 - A Graphical Approach to College Algebra

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

Step 1: Simplify the Equation for (a)

For part (a), we have the absolute value equation \(|7 - 2x| = 3\). This can be broken down into two separate equations: 1. \(7 - 2x = 3\) 2. \(7 - 2x = -3\).

Step 2: Solve Each Equation for (a)

Solve the first equation: \[7 - 2x = 3\] Subtract 7 from both sides: \[-2x = -4\] Divide by -2: \[x = 2\]Solve the second equation: \[7 - 2x = -3\] Subtract 7 from both sides: \[-2x = -10\] Divide by -2: \[x = 5\] The solutions for part (a) are \(x = 2\) and \(x = 5\).

Step 3: Set Up Inequality for (b)

For part (b), the inequality \(|7 - 2x| \geq 3\) creates two scenarios:1. \(7 - 2x \geq 3\)2. \(7 - 2x \leq -3\).

Step 4: Solve Each Inequality for (b)

For the first inequality:\[7 - 2x \geq 3\] Subtract 7 from both sides:\[-2x \geq -4\] Divide by -2 (flip the inequality sign):\[x \leq 2\]For the second inequality:\[7 - 2x \leq -3\] Subtract 7 from both sides:\[-2x \leq -10\] Divide by -2 (flip the inequality sign):\[x \geq 5\] The solution for part (b) is \(x \leq 2\) or \(x \geq 5\).

Step 5: Set Up Inequality for (c)

For part (c), the inequality \(|7 - 2x| \leq 3\) means:1. \(7 - 2x \leq 3\)2. \(7 - 2x \geq -3\).

Step 6: Solve Each Inequality for (c)

For the first inequality:\[7 - 2x \leq 3\] Subtract 7 from both sides:\[-2x \leq -4\] Divide by -2 (flip the inequality sign):\[x \geq 2\]For the second inequality:\[7 - 2x \geq -3\] Subtract 7 from both sides:\[-2x \geq -10\] Divide by -2 (flip the inequality sign):\[x \leq 5\] The solution for part (c) is \(2 \leq x \leq 5\).

Key Concepts

Solving Inequalities

Inequalities work a lot like equations, but instead of finding exact values, we're determining a range of possible solutions that satisfy the statement. In inequalities, signs like \(\geq\) or \(\leq\) are used instead of an equal sign.

When solving absolute value inequalities, the core idea is to convert them into separate inequalities without the absolute value. For example, with \(|7 - 2x| \geq 3\), you break it down into two inequalities:

• \(7 - 2x \geq 3\)
• \(7 - 2x \leq -3\)

The first inequality deals with the inner expression remaining positive or neutral, while the second one addresses it being negative.

For both inequalities, the solutions give us a combined range that represents all possible \(x\) values that make the original inequality true. If we flip an inequality sign when dividing or multiplying by a negative number, remember that the direction of our solution changes as well.

Analytical Methods

Analytical methods refer to solving problems by breaking them into simpler parts using logic and algebraic manipulation. This involves a step-by-step approach, verifying each stage to ensure accuracy without relying on guesswork or numerical approximation.

In exercises involving absolute value equations like \(|7 - 2x| = 3\), analytical methods help us create two separate situations:

• \(7 - 2x = 3\) - considering when the expression inside the absolute value is equal to the positive side
• \(7 - 2x = -3\) - considering when the expression equals the negative side

Each situation can be simplified step-by-step, allowing us to see all possible solutions methodically.

These methods ensure that all potential cases are explored thoroughly, leading to a complete set of solutions rather than missing any plausible answers.

Algebra

Algebra serves as a powerful tool, enabling us to model real-world problems through equations. Through algebra, we manipulate symbols and numbers to discover values that satisfy given mathematical statements.

When dealing with absolute value and inequalities, algebra guides us in:

• Applying operations like addition, subtraction, multiplication, and division to both sides of an equation or inequality
• Leveraging symmetry in properties such as \(|x|\) implying both \(x\) and \(-x\)
• Understanding the effects of reversing inequality signs when multiplying or dividing by a negative

Algebra not only helps in solving equations but also aids in visualizing and explaining solutions comprehensively, providing the student with a tangible understanding of abstract concepts.

Through consistent practice with algebra, one builds confidence in not just solving equations but also applying them to diverse and complex problems efficiently.