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Chapter 1: Problem 27

Solve the linear inequality. Express the solution using interval notation and graph the solution set. $$ 2 \leq x+5<4 $$

Short Answer

Expert verified
The solution is \([-3, -1)\) and can be graphed with a solid dot at \(-3\) and an open dot at \(-1\).

Step by step solution

01

Understand the Compound Inequality

The given inequality is compound: \(2 \leq x+5 < 4\). This means we have two conditions that \(x\) must satisfy simultaneously: \(2 \leq x+5\) and \(x+5 < 4\). We will solve each inequality separately.
02

Solve the First Inequality

First, we solve \(2 \leq x+5\). Subtract 5 from both sides to isolate \(x\): \[2 - 5 \leq x+5 - 5\]\[-3 \leq x\].
03

Solve the Second Inequality

Now, we solve \(x+5 < 4\). Subtract 5 from both sides to isolate \(x\): \[x+5 - 5 < 4 - 5\]\[x < -1\].
04

Combine the Solutions

The solutions to the compound inequality are both conditions satisfied simultaneously: \(-3 \leq x < -1\). This is the interval for which the inequality holds true.
05

Express the Solution in Interval Notation

In interval notation, the solution \(-3 \leq x < -1\) is expressed as \([-3, -1)\). The interval uses a square bracket \([\) for \(-3\) because it's included and a parenthesis \()\) for \(-1\) because it's not included.
06

Graph the Solution Set

To graph the solution set on a number line, draw a solid dot at \(-3\) to indicate it is included in the solution. Use an open dot at \(-1\) to show that it is not included. Shade the region between \(-3\) and \(-1\).

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Key Concepts

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

Compound Inequalities
A compound inequality is essentially two inequalities combined into one statement by using either "and" or "or". It helps us find the range of solutions for a variable that satisfies multiple conditions at the same time. In our example, we were given the inequality \(2 \leq x+5 < 4\). Here, the two inequalities \(2 \leq x+5\) and \(x+5 < 4\) must both be true for \(x\) to be a valid solution. The word "and" is implicit, meaning solutions must satisfy both conditions. Working with compound inequalities is beneficial when you need to locate a range that suits specific criteria.
Interval Notation
Interval notation is a concise way to describe a set of numbers along a number line. In this notation, a pair of numbers are written as an interval, representing all numbers between them. Based on our solved inequality \(-3 \leq x < -1\), let's see how interval notation works:
  • \([-3, -1)\) is read as "from -3 to just below -1".
  • The square bracket \([\) on \(-3\) means -3 is included in the solution.
  • The parenthesis \()\) on \(-1\) shows -1 is not included.
Utilizing interval notation in mathematics eases communication and understanding of complex solution sets.
Number Line Graphing
Graphing inequalities on a number line is a helpful visual tool to quickly identify which parts of the number line are included in an inequality's solution:
  • In our example, the solution \([-3, -1)\) is depicted by shading the area between -3 and -1.
  • A solid dot at \(-3\) indicates that -3 is part of the solution, as it satisfied \(-3 \leq x\).
  • An open dot at \(-1\) indicates -1 is not part of the solution, reflecting the "less than" part \(x < -1\).
This graphical representation allows us and others to visualize the range of possible values for \(x\) easily.
Solving Inequalities
Solving inequalities is much like solving equations but requires a keen attention to inequality signs and the rules they entail. Here's how we solve them step-by-step:
  • When tackling each inequality, isolate the variable on one side to determine its range of values.
  • For \(2 \leq x+5\), subtract 5 from both sides to get \(-3 \leq x\). This tells us \(x\) can be -3 or any larger number.
  • For \(x+5 < 4\), subtract 5 to determine \(x < -1\). This sets an upper boundary for \(x\).
  • Combine these solutions. The intersection, or overlapping part of both sets, provides the valid solution: \(-3 \leq x < -1\).
Careful application of addition, subtraction, and multiplication to both sides of an inequality helps find the correct solution, but remember if you multiply or divide by a negative number, the inequality sign flips.

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Most popular questions from this chapter

Recall that the symbol \(\overline{z}\) represents the complex conjugate of \(z .\) If \(z=a+b i\) and \(w=c+d i,\) prove each statement. $$ \overline{\overline{z}}=z $$

Evaluate the expression and write the result in the form a bi. $$ (3-2 i)+\left(-5-\frac{1}{1} i\right) $$

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Evaluate the expression and write the result in the form a bi. $$ (-4+i)-(2-5 i) $$

Evaluate the expression and write the result in the form a bi. $$ (2-5 i)+(3+4 i) $$
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