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Chapter 4: Problem 550

Graph the linear inequality \(x \leq-1\).

Short Answer

Expert verified
Graph a solid vertical line at \(x = -1\) and shade to the left.

Step by step solution

01

- Understand the Inequality

The given inequality is in the form of a linear inequality: \(x \leq -1\). This means for any value of x to be a solution, it must be less than or equal to -1.
02

- Draw the Boundary Line

The boundary line for the inequality is \(x = -1\). Draw a vertical line through \(x = -1\). Since the inequality is \(\leq \) (less than or equal to), the line will be solid.
03

- Shade the Solution Region

Shade the region to the left of the line \(x = -1\), because the inequality represents all values of x that are less than or equal to -1.

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Key Concepts

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

linear inequalities
Linear inequalities are expressions that relate linear equations using inequality symbols such as \(\textless\), \(\textless =\), \(\textgreater\), and \(\textgreater=\). These symbols are used to show that one side of the equation is not specifically equal to, but rather greater than or less than, the other side. In our exercise, we have the inequality \(x \leq -1\). This tells us that any solution must make \(x\) less than or equal to -1. Understanding what this means is crucial as it forms the foundation for both graphing and identifying the solution region.
graphing
Graphing an inequality helps visually represent all possible solutions. To start graphing, convert the inequality into an equation to find the boundary line. For \(x \leq -1\), the boundary line is \(x = -1\). Draw this line on a graph to see where your solutions lie. Since \(x = -1\) is a simple vertical line, place it where \(x\) equals -1 for all values of \(y\). Because the inequality is 'less than or equal to', this boundary line will be solid, indicating that points exactly on the line are included in the solution set. For more complex inequalities involving \(y\), breaking down the steps ensures you correctly graph each part.
solution region
The solution region shows all points that satisfy the inequality. Once the boundary line is drawn, determine which side of it contains solutions by selecting a test point not on the line. For the inequality \(x \leq -1\), choose a point to the left of \(x = -1\), such as \(x = -2\). If substituting this test point into the inequality holds true, shade this half-plane. In our case, because \(x = -2\) satisfies \(x \leq -1\), we shade everything to the left of \(x = -1\). This shaded area is the solution region where any point within it satisfies the inequality.
boundary line
The boundary line is critical in visualizing linear inequalities. When graphing \(x \leq -1\), the boundary line \(x = -1\) is drawn solid. This signifies points on the line are included in the solution. If the inequality did not have \(\leq\) or \(\geq\), the line would be dashed, showing those points are not part of the solution. The boundary line splits the graph into two areas: one that contains solutions and one that does not. Accurately drawing and interpreting this line helps correctly identify the solution region.

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Most popular questions from this chapter

Graph the linear inequality \(x>-3 .\)

Find the equation of each line. Write the equation in slope-intercept form. \(m=\frac{5}{6},\) containing point (6,7)

Find the equation of each line. Write the equation in slope-intercept form. \(m=-\frac{3}{5},\) containing point (10,-5)

Find an equation of a line perpendicular to the given line and contains the given point. Write the equation in slope-intercept form. line \(y-6=0,\) point 4(-5,-3)

Find an equation of a line perpendicular to the given line and contains the given point. Write the equation in slope-intercept form. line \(y=-x+5,\) point(3,3)
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