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Chapter 1: Problem 51

Solve each compound inequality. Write the solution set using interval notation and graph it. $$\frac{1}{2}(x+1)>3 \text { or } 0<7-x$$

Short Answer

Expert verified
The solution set is \((-\infty, 7) \cup (5, \infty)\).

Step by step solution

01

Solve the first inequality

First, solve the inequality \( \frac{1}{2}(x+1)>3 \). Start by eliminating the fraction. Multiply both sides by 2: \( x + 1 > 6 \). Next, isolate \( x \) by subtracting 1 from both sides: \( x > 5 \).
02

Solve the second inequality

Next, solve the inequality \( 0<7-x \). Start by isolating \( x \). Subtract 7 from both sides: \( -7 < -x \). Now, divide both sides by -1 (remember to reverse the inequality sign when dividing by a negative number): \( 7 > x \) or equivalently \( x < 7 \).
03

Combine the solution sets

Since the inequalities are combined with 'or,' the solution set is any \( x \) that satisfies either of them. So, the solution set is \( x > 5 \) or \( x < 7 \). This can be written in interval notation as \( (-\infty, 7) \cup (5, \infty) \).
04

Graph the solutions

Draw a number line and shade the region that represents the interval notation \((-\infty, 7) \cup (5, \infty)\). The number 7 is an open circle and 5 is also an open circle because they are not included in the solution set.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

inequality solutions
Inequalities are mathematical expressions involving the symbols >, <, 鈮�, or 鈮�. Solving inequalities means finding the set of values that make the expression true.
For example, solving the inequality \( \frac{1}{2}(x+1)>3 \) means finding all the values of \( x \) that make the inequality true. Let's break down the steps:
  • Eliminate fractions: Multiply both sides by 2.
  • Isolate the variable: Subtract 1 from both sides.
Thus, solving \( \frac{1}{2}(x+1)>3 \) gives us \( x>5 \).
For compound inequalities like \( \frac{1}{2}(x+1)>3 \text{ or } 0<7-x \), solve each part separately and then combine the solutions. Inequalities joined by 'or' require that either condition is met. Remember:
  • For inequalities with 'and,' both conditions must be true.
  • For inequalities with 'or,' either condition can be true.
interval notation
Interval notation provides a way to describe sets of numbers. It's especially useful for expressing solutions to inequalities. For example, the solution for \( x>5 \) or \( x<7 \) is written in interval notation as \( (-\infty, 7) \cup (5, \infty) \). Here are some key points to remember about interval notation:
  • Use parentheses \( () \) for open intervals where the endpoint is not included.
  • Use square brackets \( [] \) for closed intervals where the endpoint is included.
  • The union symbol \( \cup \) is used to combine disjoint intervals.
Let's dissect our example:
\( (-\infty, 7) \cup (5, \infty) \) means all values less than 7 or all values greater than 5. Note that 7 and 5 are not included because the inequalities are strict \( < \) and \( > \).
graphical representation of inequalities
Graphing inequalities helps visually understand the solution sets. Let's graph the solution for the given compound inequality, which is \((-\infty,7) \cup (5, \infty)) \).
Here鈥檚 how to graph it:
  • Draw a number line.
  • Identify critical points, which are 5 and 7 in this case.
  • Use open circles to denote that 5 and 7 are not included in the solution set.
  • Shade the regions to the left of 7 and to the right of 5.
The open circles at 5 and 7 indicate that those numbers themselves are not part of the solution. Shading the regions correctly shows the infinite span of allowable values. Graphically representing the inequality helps you see the range of values that satisfy the inequality conditions.

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