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Chapter 8: Problem 76

Solve and graph each solution set. $$ x+5<-3 \text { or } x+5 \geq 4 $$

Short Answer

Expert verified
The solution set is \( (-\infty, -8) \cup [-1, \infty) \) . Graph the intervals as described.

Step by step solution

01

Solve the first inequality

The first inequality given is \( x + 5 < -3 \). Subtract 5 from both sides to isolate \( x \): \[ x + 5 - 5 < -3 - 5 \] Simplifying, \[ x < -8 \].
02

Solve the second inequality

The second inequality given is \( x + 5 \geq 4\). Subtract 5 from both sides to isolate \( x \): \[ x + 5 - 5 \geq 4 - 5 \] Simplifying, \[ x \geq -1 \].
03

Combine the solutions

The solutions for the inequalities are \( x < -8 \) and \( x \geq -1 \) . The solution set is the union of the two intervals: \( (-\infty, -8) \cup [-1, \infty) \).
04

Graph the solution set

The solution set includes all points less than -8 and all points greater than or equal to -1. On the number line, this can be represented as: - Draw an open circle at -8 to denote that -8 is not included in the solution set and shade to the left. - Draw a closed circle at -1 to denote that -1 is included in the solution set and shade to the right. The graph will show two distinct portions.

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Key Concepts

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

solving inequalities
When tackling an inequality such as \( x + 5 < -3 \) or \( x + 5 \geq 4 \), the first step is to solve each inequality separately.
The goal is to isolate the variable \( x \) on one side of the inequality sign.
For example, let's start with the first inequality:
  • For \( x + 5 < -3 \), subtract 5 from both sides to get \( x < -8 \).
  • For the second inequality, \( x + 5 \geq 4 \), also subtract 5 from both sides to isolate \( x \), resulting in \( x \geq -1 \).
These steps will give us two separate solutions for \( x \), one for each inequality.
union of intervals
Combining the solutions of the inequalities involves taking the union of both solution sets. This means we are including all values that satisfy either of the inequalities.
For our example:
  • The solution set for \( x + 5 < -3 \) is expressed as the interval \( (-\infty, -8) \).
  • The solution set for \( x + 5 \geq 4 \) is expressed as the interval \( [-1, \infty) \).
The union of these two intervals, represented as \( (-\infty, -8) \cup [-1, \infty) \), encompasses all values that satisfy either \( x < -8 \) or \( x \geq -1 \).
Essentially, union of intervals allows us to combine the results of multiple inequalities into a single solution set.
graphing solutions
Graphing the solution sets on a number line is a helpful way to visualize the intervals. This is a straightforward process:
  • First, draw an open circle at -8 to indicate that -8 itself is not included in the solution (for \( x < -8 \)). From there, shade the number line to the left indicating all values less than -8.
  • Next, draw a closed circle at -1 to show that -1 is included in the solution set (for \( x \geq -1 \)). From that point, shade the number line to the right to indicate all values greater than or equal to -1.
The graph will show two separate portions:
The left part shaded including all values less than -8, and the right part shaded including all values greater than or equal to -1.
This graphical representation helps in quickly identifying the span of values that are part of the solution set.

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Most popular questions from this chapter

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The function given by \(w(t)=0.0125 t+4.525\) can be used to estimate the number of pounds of solid waste, \(w(t),\) produced daily, on average, by each person in the United States, \(t\) years after \(2000 .\) For what years will waste production range from 4.6 lb to 4.8 lb per person per day?

Find an equivalent inequality with absolute value. $$ x<-8 \text { or } 2

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