Problem 65
Question
Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The graph of \(x<4\) is the half-plane to the left of the vertical line described by \(x=4\)
Step-by-Step Solution
Verified Answer
The statement is true, as the graph of the inequality \(x<4\) indeed represents the half-plane to the left of the vertical line \(x=4\).
1Step 1 Analyze the statement
The first step in tackling this exercise is to understand the statement given. The inequality \(x<4\) translates to 'x is less than 4'. This suggests that all values satisfying the inequality are less than four. In graphing this, it is expected to have all points to the left of the line \(x=4\), as lesser values are located on the left in a number line. Thus, from this analysis, the statement seems to be true.
2Step 2 Compare with the graphical rule
The next step is to compare the graphical rule of linear inequalities. This rule states that for any \(x
3Step 3 Conclude the validation
Based on the rule of graphical representation and comparison with the statement, it's clear that the statement, 'The graph of \(x<4\) is the half-plane to the left of the vertical line described by \(x=4\)', is true.
Key Concepts
Graphing InequalitiesLinear InequalitiesHalf-Plane
Graphing Inequalities
Graphing inequalities involves plotting the solution set of an inequality on a coordinate plane. For linear inequalities, this typically means shading a region that represents all potential solutions.
In the case of a simple inequality like \(x < 4\), the solution involves plotting a vertical line at \(x = 4\) and shading the half-plane that lies to the left of this line.
The line itself is usually represented as dashed to indicate that points on the line are not included in the solution set (since we are dealing with a strict inequality with no equal sign). This visual representation allows for quick identification of all possible solutions of the inequality.
In the case of a simple inequality like \(x < 4\), the solution involves plotting a vertical line at \(x = 4\) and shading the half-plane that lies to the left of this line.
The line itself is usually represented as dashed to indicate that points on the line are not included in the solution set (since we are dealing with a strict inequality with no equal sign). This visual representation allows for quick identification of all possible solutions of the inequality.
Linear Inequalities
Linear inequalities are similar to linear equations, but instead of depicting equal relationships, they describe ranges of possible solutions.
A linear inequality looks much like a linear equation but includes inequality signs such as \(<\), \(>\), \(\leq\), or \(\geq\) instead of \(=\).
Consider the inequality \(x < 4\). It represents all numbers less than 4, without including 4 itself. Therefore, in a linear system expressed graphically, you only shade the region where all the possible values fall below this threshold.
A linear inequality looks much like a linear equation but includes inequality signs such as \(<\), \(>\), \(\leq\), or \(\geq\) instead of \(=\).
Consider the inequality \(x < 4\). It represents all numbers less than 4, without including 4 itself. Therefore, in a linear system expressed graphically, you only shade the region where all the possible values fall below this threshold.
- The inequality sign specifies whether you shade above or below the line.
- The line associated with the inequality is dashed if the inequality is strict (\(<\) or \(>\)).
Half-Plane
In graphing, a half-plane is a region of the coordinate plane on one side of a line. This line acts as a boundary that helps define the solutions of a linear inequality.
When graphing the inequality \(x < 4\), we draw a vertical line at \(x = 4\) and shade the area to its left.
The shaded region, or half-plane, includes all the \(x\) values that satisfy the inequality. Keep in mind that the boundary line itself does not belong to the solution if the inequality is strict.
The concept of a half-plane is a visual method, making it easier for students to understand and interpret solutions of linear inequalities through simple observation of a graph. This visual aid not only enhances comprehension but also facilitates problem-solving when dealing with multiple inequalities.
When graphing the inequality \(x < 4\), we draw a vertical line at \(x = 4\) and shade the area to its left.
The shaded region, or half-plane, includes all the \(x\) values that satisfy the inequality. Keep in mind that the boundary line itself does not belong to the solution if the inequality is strict.
The concept of a half-plane is a visual method, making it easier for students to understand and interpret solutions of linear inequalities through simple observation of a graph. This visual aid not only enhances comprehension but also facilitates problem-solving when dealing with multiple inequalities.
Other exercises in this chapter
Problem 64
Write an equation in the form \(y=m x+b\) of the line that is described. The line falls from left to right. It passes through the origin and a second point with
View solution Problem 64
In Exercises \(61-64\), determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
View solution Problem 65
graph each linear equation in two variables. Find at least five solutions in your table of values for each equation. $$y=-3 x-1$$
View solution Problem 66
graph each linear equation in two variables. Find at least five solutions in your table of values for each equation. $$y=-3 x-2$$
View solution