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Chapter 8: Problem 71

Solve and graph each solution set. $$ 4

Short Answer

Expert verified
The solution set is \(0 < x < 4\).

Step by step solution

01

Solve the first inequality

Start with the first inequality \(4 < x + 4\). Subtract 4 from both sides to isolate \(x\): \[4 - 4 < x + 4 - 4\] \[0 < x\] or \[x > 0\]
02

Solve the second inequality

Next, consider the second inequality \(x - 1 < 3\). Add 1 to both sides to isolate \(x\): \[x - 1 + 1 < 3 + 1\] \[x < 4\]
03

Combine the solutions

Combine the solutions from Step 1 and Step 2. \[0 < x < 4\] The solution is the intersection of both inequalities.
04

Graph the solution set on a number line

Draw a number line. Mark open circles at 0 and 4 since the inequalities are strictly greater than and less than. Shade the region between 0 and 4 to represent all values of \(x\).

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Key Concepts

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

Understanding Inequalities
Inequalities are mathematical statements that compare two expressions using inequality symbols like <, >, 鈮�, or 鈮�. Unlike equations, which show that two expressions are equal, inequalities show that one expression is either less than or greater than the other. Here, we are dealing with compound inequalities. This means we have more than one inequality to solve, and the solution must satisfy all parts.

In the exercise:
  • For the first part, 4 < x + 4, we need to find all values of x that make this statement true.
  • For the second part, x - 1 < 3, we need to find all values of x that satisfy this condition.
Understanding how to isolate the variable in each part of the compound inequality will help you solve the whole problem correctly.
Finding the Solution Set
The solution set is a range of values that satisfy the given inequalities. Let's break down how we found the solution set step by step:

  • First, we tackled the inequality 4 < x + 4. By subtracting 4 from both sides, we isolated x and got 0 < x, which means x > 0.
  • Next, we worked on the inequality x - 1 < 3. Adding 1 to both sides, we isolated x to get x < 4.
  • To find the intersection of these two solutions, we combined them. This gave us 0 < x < 4, meaning x can be any value greater than 0 and less than 4.
By solving each inequality separately and then combining them, we determine the complete solution set.
Graphing Inequalities
Graphing inequalities on a number line is a great way to visualize the solution set. Here's how to graph the solution 0 < x < 4 step by step:

  • Start by drawing a horizontal number line.
  • Mark an open circle at 0 and another at 4 to show that these values are not included in the solution set.
  • Shade the region between 0 and 4 to represent all the values that x can take (greater than 0 but less than 4).
This shaded region is your graphical representation of the solution. It visually shows that any point between 0 and 4 is included in the solution set but not the endpoints themselves. Graphing helps to ensure you understand which numbers work in the inequalities.

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