Problem 37
Question
Use interval notation to express solution sets and graph each solution set on a number line. Solve linear inequality. \(2 x-11<-3(x+2)\)
Step-by-Step Solution
Verified Answer
The solution to the inequality is \(x<1\), expressed in interval notation as \((-\infty, 1)\). The graph of this solution on a number line consists of a circle at 1 (not filled in) and an arrow to the left from the circle.
1Step 1: Solve the Inequality
Start by distributing the -3 on the right side of the inequality: \(2x - 11 < -3x - 6\). Then move the x terms to one side and the constant terms to the other side to get: \(5x < 5\). Finally, divide both sides by 5 to solve for \(x\), which gives: \(x<1\)
2Step 2: Express in Interval Notation
Interval notation for \(x < 1\) is \((-\infty, 1)\). This represents all values from negative infinity up to, but not including, 1.
3Step 3: Graph on a Number Line
To graph this interval on a number line, draw a number line and put a circle at 1 to represent that 1 is not included in the interval. Then draw an arrow to the left from the circle to represent all values less than 1 on the number line.
Key Concepts
Interval NotationNumber Line GraphingLinear Algebra
Interval Notation
Grasping the essence of interval notation is pivotal in communicating solutions to inequalities in a precise and concise manner. Think of interval notation as a simple language used to describe intervals on a real number line. It eliminates the ambiguity associated with traditional inequality symbols and enables a more streamlined representation of number sets.
When we say that the solution to an inequality is represented by \(x < 1\), interval notation transforms this statement into \( (-\infty, 1) \). Here, the parentheses indicate that the endpoint, 1, is not part of the solution. It's like saying, 'start from beyond the lowest depths (-\infty) and go all the way up to just before the number 1'. If 1 were included, we would use a square bracket instead [1]. Similarly, the symbol '-\infty' symbolizes that we're considering all numbers less to the left, without bound.
Making use of interval notation is especially beneficial when dealing with multiple intersecting or combining intervals. By understanding this system, students can not only solve inequalities but can easily communicate complex numerical ideas.
When we say that the solution to an inequality is represented by \(x < 1\), interval notation transforms this statement into \( (-\infty, 1) \). Here, the parentheses indicate that the endpoint, 1, is not part of the solution. It's like saying, 'start from beyond the lowest depths (-\infty) and go all the way up to just before the number 1'. If 1 were included, we would use a square bracket instead [1]. Similarly, the symbol '-\infty' symbolizes that we're considering all numbers less to the left, without bound.
Making use of interval notation is especially beneficial when dealing with multiple intersecting or combining intervals. By understanding this system, students can not only solve inequalities but can easily communicate complex numerical ideas.
Number Line Graphing
With the solution to an inequality in hand, the next step is to visually represent it on a number line. This is where number line graphing plays a significant role—it helps in visualizing the range of possible solutions.
The process is much like sketching a timeline. For the inequality \(x < 1\), you'll start by drawing a horizontal line. Then, pinpoint where the number 1 falls on this line. Instead of a dot, which would imply that 1 is included, we mark it with an open circle, reinforcing that 1 is an endpoint that the solution approaches but never reaches. To depict all values less than 1, an arrow is drawn extending leftward from the open circle. It's as though the open circle is saying 'everything up to here' and the arrow is saying 'keep going indefinitely in this direction'.
This graphical approach is not just a depiction; it's a powerful tool to immediately understand and interpret inequalities. Through practice, students can quickly start to 'see' the solutions to problems by recognizing these graphical shortcuts.
The process is much like sketching a timeline. For the inequality \(x < 1\), you'll start by drawing a horizontal line. Then, pinpoint where the number 1 falls on this line. Instead of a dot, which would imply that 1 is included, we mark it with an open circle, reinforcing that 1 is an endpoint that the solution approaches but never reaches. To depict all values less than 1, an arrow is drawn extending leftward from the open circle. It's as though the open circle is saying 'everything up to here' and the arrow is saying 'keep going indefinitely in this direction'.
This graphical approach is not just a depiction; it's a powerful tool to immediately understand and interpret inequalities. Through practice, students can quickly start to 'see' the solutions to problems by recognizing these graphical shortcuts.
Linear Algebra
The study of linear algebra provides tools for solving equations and inequalities involving variables. While linear algebra can explore much more complex relationships in higher dimensions, it begins with understanding lines and their equations in a two-dimensional space.
In the given problem, \(2x - 11 < -3(x + 2)\), we engage in the process of algebraic manipulation to find the value(s) for which this inequality holds true. This process includes distributing, combining like terms, and finally finding the solution by isolating the variable. The result, \(x < 1\), not only serves as an answer to the inequality but also as a foundational concept for graphing linear functions and understanding their behavior.
Recognizing that algebraic solutions can be represented graphically connects students to the practical applications of linear algebra. Many real-world problems in fields such as engineering, economics, and the sciences rely on this branch of mathematics for modeling relationships and predicting outcomes. A proficiency in linear algebra, therefore, unlocks a deeper comprehension of how numerical relationships can describe and define the world around us.
In the given problem, \(2x - 11 < -3(x + 2)\), we engage in the process of algebraic manipulation to find the value(s) for which this inequality holds true. This process includes distributing, combining like terms, and finally finding the solution by isolating the variable. The result, \(x < 1\), not only serves as an answer to the inequality but also as a foundational concept for graphing linear functions and understanding their behavior.
Recognizing that algebraic solutions can be represented graphically connects students to the practical applications of linear algebra. Many real-world problems in fields such as engineering, economics, and the sciences rely on this branch of mathematics for modeling relationships and predicting outcomes. A proficiency in linear algebra, therefore, unlocks a deeper comprehension of how numerical relationships can describe and define the world around us.
Other exercises in this chapter
Problem 36
Perform the indicated operations and write the result in standard form. $$ (-2+\sqrt{-11})^{2} $$
View solution Problem 36
Solve each equation with rational exponents. Check all proposed solutions. $$8 x^{\frac{5}{3}}-24-0$$
View solution Problem 37
contain rational equations with variables in denominators. For each equation, a. write the value or values of the variable that make a denominator zero. These a
View solution Problem 37
Exercises \(37-38\) involve markup, the amount added to the dealer's cost of an item to arrive at the selling price of that item. The selling price of a refrige
View solution