Problem 64
Question
Solve the inequality. Then graph its solution. $$-5<-x \leq-1$$
Step-by-Step Solution
Verified Answer
The solutions to the inequality -5<-x≤-1 are x>-5 and x≤1
1Step 1: Understand the inequality
In this exercise, the inequality is -5<-x≤-1. This inequality states that -x is greater than -5 but less than or equal to -1.
2Step 2: Solve for x
The inequality -5<-x≤-1 is equivalent to -5
3Step 3: Graphing
To graph the solution set, a number line is created. An open circle is placed at -5 and a closed circle at 1. Because x is less than -5 and greater than or equal to 1, an arrow is drawn from -5 to the left, and a line is drawn from 1 to the right. This visually illustrates the solution set of the inequality.
Key Concepts
Inequality GraphingReversing InequalityInequality Notation
Inequality Graphing
Inequality graphing is a visual representation of the solutions to an inequality on a number line or coordinate plane. It's particularly helpful in understanding the range of numbers that satisfy an inequality.
For example, when working with the inequality \[\begin{equation}-5 < -x \text{ and } -x \text{ and } -x \leq -1\end{equation}\], the graphing step involves placing an open circle at -5 to show that -5 is not included in the solution set (since the inequality is strict, '<') and a closed circle at 1 to indicate that 1 is included (represented by '\leq' which means 'less than or equal to').
To illustrate this on a number line, draw a line with an arrow extending to the left from the open circle at -5 which shows all the numbers smaller than -5 are included. Similarly, from the closed circle at 1, a line with an arrow to the right indicates that all numbers greater than or equal to 1 are solutions to the inequality. This graphic representation makes it easy to see the set of numbers that are solutions to the inequality.
For example, when working with the inequality \[\begin{equation}-5 < -x \text{ and } -x \text{ and } -x \leq -1\end{equation}\], the graphing step involves placing an open circle at -5 to show that -5 is not included in the solution set (since the inequality is strict, '<') and a closed circle at 1 to indicate that 1 is included (represented by '\leq' which means 'less than or equal to').
To illustrate this on a number line, draw a line with an arrow extending to the left from the open circle at -5 which shows all the numbers smaller than -5 are included. Similarly, from the closed circle at 1, a line with an arrow to the right indicates that all numbers greater than or equal to 1 are solutions to the inequality. This graphic representation makes it easy to see the set of numbers that are solutions to the inequality.
Reversing Inequality
Reversing inequality is an important concept that comes into play when dealing with inequalities involving multiplication or division by a negative number. Whenever you multiply or divide both sides of an inequality by a negative number, the inequality sign must be flipped in the opposite direction.
For our given problem, we started with \[\begin{equation}-5 < -x\leq -1\end{equation}\]. To find the solution for \(x\), we multiply by -1 and as a result, the inequality signs switch direction to become \[\begin{equation}-5 < x\geq 1\end{equation}\]. This reversal is critical because neglecting to reverse the inequality would result in an incorrect solution set. It ensures that the relative order of the numbers is preserved given the properties of inequalities with negative numbers.
For our given problem, we started with \[\begin{equation}-5 < -x\leq -1\end{equation}\]. To find the solution for \(x\), we multiply by -1 and as a result, the inequality signs switch direction to become \[\begin{equation}-5 < x\geq 1\end{equation}\]. This reversal is critical because neglecting to reverse the inequality would result in an incorrect solution set. It ensures that the relative order of the numbers is preserved given the properties of inequalities with negative numbers.
Inequality Notation
Inequality notation expresses the relationship between numbers or expressions with regard to their size. It tells us whether one number is greater than, less than, or equal to another number and whether or not the number in question is included in the set of solutions.
In the inequality \[\begin{equation}-5 < -x\leq -1\end{equation}\], the '<' symbol represents a strict inequality, indicating that the number on the left is strictly less than the number on the right, but is not included as a solution. On the other hand, the '\leq' symbol represents a non-strict inequality, meaning that -1 is both less than and included as a possible solution to the inequality expressed by \(x\). Understanding and using the correct inequalities symbols is crucial for both solving inequalities accurately and representing their solutions on a number line.
In the inequality \[\begin{equation}-5 < -x\leq -1\end{equation}\], the '<' symbol represents a strict inequality, indicating that the number on the left is strictly less than the number on the right, but is not included as a solution. On the other hand, the '\leq' symbol represents a non-strict inequality, meaning that -1 is both less than and included as a possible solution to the inequality expressed by \(x\). Understanding and using the correct inequalities symbols is crucial for both solving inequalities accurately and representing their solutions on a number line.
Other exercises in this chapter
Problem 63
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