Problem 32 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find all the values of \(x\) that satisfy at least one of the two inequalities. (a) \(2 x-7>1\) or \(2 x+1<3\) (b) \(2 x-7 \leq 1\) or \(2 x+1<3\) (c) \(2 x-7 \leq 1\) or \(2 x+1>3\)

Step-by-Step Solution

Answer

(a) x < 1 or x > 4; (b) x ≤ 4; (c) all x.

Step 1: Solve the first inequality for (a)

Solve the inequality \(2x - 7 > 1\):\[2x - 7 > 1 \2x > 8 \x > 4\]This means \(x\) must be greater than 4.

Step 2: Solve the second inequality for (a)

Solve the inequality \(2x + 1 < 3\):\[2x + 1 < 3 \2x < 2 \x < 1\]This means \(x\) must be less than 1.

Step 3: Combine solutions for (a)

Since the solutions must satisfy "or," find the union of \(x > 4\) and \(x < 1\):- \(x < 1\) or \(x > 4\).Thus, this includes all values except \(1 \le x \le 4\).

Step 4: Solve the first inequality for (b)

Solve the inequality \(2x - 7 \leq 1\):\[2x - 7 \leq 1 \2x \leq 8 \x \leq 4\]This means \(x\) must be less than or equal to 4.

Step 5: Solve the second inequality for (b)

Solve the inequality \(2x + 1 < 3\):\[2x + 1 < 3 \2x < 2 \x < 1\]This means \(x\) must be less than 1.

Step 6: Combine solutions for (b)

Since the solutions must satisfy "or," find the union of \(x \leq 4\) and \(x < 1\):- \(x \leq 4\)Thus, the solution is \(x \leq 4\) because it covers both inequalities.

Step 7: Solve the first inequality for (c)

Solve the inequality \(2x - 7 \leq 1\):\[2x - 7 \leq 1 \2x \leq 8 \x \leq 4\]This means \(x\) must be less than or equal to 4.

Step 8: Solve the second inequality for (c)

Solve the inequality \(2x + 1 > 3\):\[2x + 1 > 3 \2x > 2 \x > 1\]This means \(x\) must be greater than 1.

Step 9: Combine solutions for (c)

Since the solutions must satisfy "or," find the union of \(x \leq 4\) and \(x > 1\):- \(x > 1\) or \(x \leq 4\).Thus, the solution is all \(x\) since every number satisfies at least one inequality.

Key Concepts

Inequality Solving

Inequalities are a crucial part of algebra that express the relationship between expressions that are not equal. Solving inequalities involves finding the entire set of values for a variable that satisfy the inequality condition. Let's break this down with an example: consider the inequality \(2x - 7 > 1\). To solve it, we need to isolate \(x\):

• First, add 7 to both sides: \(2x > 8\).
• Next, divide each side by 2: \(x > 4\).

This process is about simplifying the inequality step by step until the variable is isolated. Note, in inequalities, flipping the inequality sign is necessary when multiplying or dividing both sides by a negative number. However, in our example, that step wasn't required. Understanding each algebraic step ensures the integrity of the inequality remains intact, leading us to the correct set of solutions.

Mathematical Reasoning

Mathematical reasoning involves applying logical thought processes to solve problems. In inequality problems, reasoning helps determine the nature of the relationship between variables and assists in interpreting results. For instance, in a situation involving "or" in inequalities such as \(x > 4\) or \(x < 1\), it doesn't mean both conditions must be true simultaneously. Instead, a value of \(x\) needs to satisfy at least one condition. This logical operation, known as a disjunction, indicates flexibility in solutions because the condition is less restrictive. Utilizing reasoning to understand these conditions aids in effectively transitioning from the algebraic solution to what it practically means for variable values. This logical aspect helps learn not only solving techniques but also how to interpret results comprehensively in real-world contexts.

Union of Solutions

The union of solutions is a method used when dealing with more than one inequality. It involves combining the sets of solutions that satisfy each inequality. For scenario (a), where we have two inequalities: \(x > 4\) and \(x < 1\), their union is symbolic of all values that meet either condition. Here’s how:

• Identify solutions for each inequality individually.
• Combine the solutions to form a union where the solutions of either are covered: \(x < 1\) or \(x > 4\).

This results in a series of numbers that exclude those in the range \([1,4]\), illustrating values outside this interval satisfy at least one of the inequalities. The union is powerful in inequalities, providing a comprehensive set of possible solutions that acknowledge overlaps yet respect the logical constraints of each inequality. Through this union process, students can gain a holistic view of solution sets, shedding light on the interrelationships between different inequality conditions.