Problem 6 - Fundamentals of Mathematics

Question

What integers can replace \(x\) so that the following statement is true? \(-5 \leq x<2\)

Step-by-Step Solution

Answer

The integers are \(-5, -4, -3, -2, -1, 0,\) and \(1\).

Step 1: Understand the Inequality

The inequality states that the values of \(x\) should be greater than or equal to \(-5\), and less than \(2\). This can be written as \(-5 \leq x < 2\).

Step 2: Identify Possible Integer Values

Identify integers that satisfy \(-5 \leq x\) and are also less than \(2\). These integers are \(-5, -4, -3, -2, -1, 0,\) and \(1\).

Step 3: Verify the Values

Check each integer to ensure they fulfill the conditions: \(-5\) is greater than or equal to \(-5\) and less than \(2\); same for \(-4, -3, -2, -1, 0, 1\). All these values satisfy the inequality as required.

Key Concepts

Number Line

A number line is a visual representation of numbers arranged in order along a straight line. It helps to easily understand the position and relation of integers and other values. When dealing with inequalities, the number line is a great tool. It allows you to visualize which numbers or integers fulfill a certain condition.

For instance, if you need to find values that satisfy \(-5 \leq x < 2\), a number line shows where these values lie. Start at \(-5\), which is included in the solution because of the 'less than or equal to' symbol (\(\leq\)). Move right up to \(2\), not included since it's only 'less than'. This span on the number line helps confirm integer solutions neatly and efficiently.

Inequality Solutions

When solving inequality statements like \(-5 \leq x < 2\), the goal is to find all integer values that make the statement true. Inequality solutions can be graphically represented using a number line, but also explicitly listed. The challenge is to remember the listing of integers that satisfy both parts of the inequality.

First, look at \(-5 \leq x\). This means \(x\) should be any integer greater than or equal to \(-5\). Next, the condition \(x < 2\) tells us that \(x\) should be less than \(2\). Combine these results by choosing integers from \(-5\) to \(1\) inclusive. These integers, \(-5, -4, -3, -2, -1, 0, 1\), satisfy the original inequality.

Integer Range

An integer range is a series of consecutive integers lying between two specified boundaries. In this context, it includes all integers that satisfy a given inequality. When dealing with the inequality \(-5 \leq x < 2\), understanding the integer range is crucial.

This range \([-5, 2)\) includes all integers starting from \(-5\) up to but not including \(2\).

• Start from the lowest boundary, \(-5\), since it's included.
  
• Move up to the point just before the upper boundary, \(2\), without including it.
  
• The integer range that fits this condition is: \(-5, -4, -3, -2, -1, 0, 1\).
  

This way, both the lower and upper boundaries of the range are respected as dictated by the inequality.

Inequality Notation

Inequality notation is a symbolic way to show the relationship between numbers where they are not equal. Symbols like \(\leq\) and \(\) help in defining the range of possible solutions. In the inequality \(-5 \leq x < 2\), two symbols are used.

• \(\leq\) stands for 'less than or equal to', meaning the number \(x\) can be exactly \(-5\) or greater.
  
• \(\) stands for 'less than', meaning \(x\) must strictly be less than \(2\).
  

This notation is very important as it informs you exactly which values qualify for the inequality statement. Interpret these symbols properly to construct the correct solution set.