Problem 31 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find all values of \(x\) that satisfy both inequalities simultaneously. (a) \(3 x+7>1\) and \(2 x+1<3\) (b) \(3 x+7>1\) and \(2 x+1>-4\) (c) \(3 x+7>1\) and \(2 x+1<-4\)

Step-by-Step Solution

Answer

(a) \(x \in (-2, 1)\), (b) \(x > -2\), (c) no solution.

Step 1: Solve the first inequality \\(3x + 7 > 1\\)

To solve the inequality \(3x + 7 > 1\), subtract 7 from both sides to isolate the term with \(x\). You get \(3x > -6\). Now, divide both sides by 3 to solve for \(x\): \(x > -2\).

Step 2: Solve the second inequality \\(2x + 1 < 3\\)

For the inequality \(2x + 1 < 3\), start by subtracting 1 from both sides to obtain \(2x < 2\). Then, divide both sides by 2, which gives \(x < 1\).

Step 3: Find the intersection of the solutions for inequalities (a)

Combine the results from Step 1 and Step 2 for part (a). We need the values of \(x\) that satisfy both \(x > -2\) and \(x < 1\). The solution is the interval \((-2, 1)\).

Step 4: Solve the second inequality \\(2x + 1 > -4\\)

For part (b), we solve the inequality \(2x + 1 > -4\). Subtract 1 from both sides, resulting in \(2x > -5\). Next, divide by 2 to find \(x > -2.5\).

Step 5: Find the intersection of the solutions for inequalities (b)

For part (b), combine \(x > -2\) (from Step 1) and \(x > -2.5\) (from Step 4). The more restrictive condition is \(x > -2\). Thus, the intersection of the solutions is \(x > -2\).

Step 6: Solve the second inequality \\(2x + 1 < -4\\)

For part (c), solve the inequality \(2x + 1 < -4\). Subtract 1 from both sides to obtain \(2x < -5\). Then, divide by 2, resulting in \(x < -2.5\).

Step 7: Find the intersection of the solutions for inequalities (c)

For part (c), to satisfy both \(x > -2\) (from Step 1) and \(x < -2.5\), there are no common values, as \(x > -2\) cannot be less than \(-2.5\). Hence, there is no solution.

Key Concepts

Understanding Algebra in Inequalities

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating these symbols. In the context of inequalities, algebra helps us express relationships between variables using symbols like "<" and ">." Let's simplify it here.
When solving an inequality, the goal is to isolate the variable, often denoted by \(x\), on one side of the inequality. This process is similar to solving equations, where you perform operations to both sides to maintain balance. However, there's a twist.
For inequalities, any time you multiply or divide both sides by a negative number, the inequality sign flips. So, \(3x + 7 > 1\) becomes \(3x > -6\), and further simplifies to \(x > -2\) after dividing both sides by 3. This step is fundamental to understand as it helps achieve a clear range of values for \(x\) that satisfy the inequality condition.

Solving and Combining Inequalities

Inequality solving involves finding solutions that satisfy certain conditions stated in inequality expressions. It's crucial to note that inequalities do not just yield single values like equations but provide a range of possible values.
To solve the inequality \(2x + 1 < 3\), subtract 1 from both sides to get \(2x < 2\). Then, divide by 2 to find \(x < 1\). You've identified a condition for the solution set of \(x\).
To find overlapping solutions for two inequalities, you look for the intersection—the common solution space—of their individual solutions. For instance, finding \((x > -2)\) and \((x < 1)\), means finding the range where both conditions are true simultaneously, which is the interval \((-2, 1)\). This concept of intersections is essential as it indicates the shared solution.

Working with Mathematical Intervals

Mathematical intervals provide a concise way to describe a range of values. Inequalities often involve the use of intervals to illustrate solution sets. Consider the solutions \(x > -2\) and \(x < 1\). The intersection of these is represented by the interval \((-2, 1)\).
This interval notation means that \(x\) can take any value greater than -2 and less than 1. Important to note, this doesn't include the endpoints -2 or 1, as indicated by the parentheses "()", which denote an open interval.
However, in more general contexts, brackets "[]" would signify closed intervals, which do include endpoints. Understanding how to read and write intervals helps interpret solution sets clearly and convey the set of numbers included in a solution.