Linear Inequalities in Two Variables: Videos & Practice Problems

Understanding linear inequalities in two variables involves graphing the corresponding line, either solid for ≤ or ≥, or dashed for < or >, and shading the region where the inequality holds true. Testing points by substituting $x$ and $y$ values confirms if they satisfy the inequality, indicating solutions within the shaded area. This approach extends from one-dimensional inequalities to multivariable polynomials, emphasizing the importance of recognizing terms, coefficients, and the degree of a polynomial to accurately interpret and graph inequalities in standard form.

Intro to Linear Inequalities in Two Variables

Understanding solutions to linear inequalities in two variables builds on the concept of ordered pairs as solutions to two-variable equations. An ordered pair, written as (x, y), is a solution to an equation if substituting the values of x and y into the equation results in a true statement. For example, if plugging in (4, -1) into an equation yields a true equality like 5 = 5, then that ordered pair lies on the line represented by the equation.

When dealing with linear inequalities, the process is similar but with an important difference. Instead of an equation like ax + by = c, inequalities take the form ax + by < c, ax + by ≤ c, ax + by > c, or ax + by ≥ c. To determine if an ordered pair is a solution to the inequality, substitute the x and y values into the inequality. If the resulting statement is true, then the ordered pair is a solution.

For instance, consider the inequality 2x + 3y < 5. Testing the point (-1, 0) involves calculating:

\[2(-1) + 3(0) = -2 + 0 = -2\]

Since -2 < 5 is true, (-1, 0) is a solution to the inequality. Graphically, solutions to inequalities are represented by a shaded region rather than just a line. This region includes all points that satisfy the inequality. The point (-1, 0) lies within this shaded area, confirming it as a solution.

In contrast, testing the point (5, 3) gives:

\[2(5) + 3(3) = 10 + 9 = 19\]

Since 19 < 5 is false, (5, 3) is not a solution and lies outside the shaded region on the graph.

It is crucial to pay attention to the inequality symbol. If the inequality is strict (e.g., < or >), the boundary line itself is not included in the solution set, so points on the line are not solutions. However, if the inequality includes equality (e.g., ≤ or ≥), then points on the line satisfy the inequality and are part of the solution set.

In summary, solving linear inequalities in two variables involves testing ordered pairs by substitution and interpreting the results graphically as regions rather than just lines. This approach helps visualize all possible solutions and understand the difference between equations and inequalities in two variables.

Given the linear inequality $4x+\frac{2}{3}y\ge 6$, determine if the following ordered pair is a solution to the inequality.
$(2,3)$

• A. Yes
• B. No
• C. Cannot be determined

Answer: (A)

Given the linear inequality $4x+\frac{2}{3}y\ge 6$, determine if the following ordered pair is a solution to the inequality.
$(-2,9)$

• A. Yes
• B. No
• C. Cannot be determined

Answer: (B)

Given the linear inequality $4x+\frac{2}{3}y\ge 6$, determine if the following ordered pair is a solution to the inequality.
$(0,9)$

• A. Yes
• B. No
• C. Cannot be determined

Answer: (A)

Given the linear inequality , determine if the following ordered pair is a solution to the inequality.
$(0,-2)$

• A. Yes
• B. No
• C. Cannot be determined

Answer: (A)

Given the linear inequality , determine if the following ordered pair is a solution to the inequality.
$(3,\frac{3}{4})$

• A. Yes
• B. No
• C. Cannot be determined

Answer: (A)

Given the linear inequality , determine if the following ordered pair is a solution to the inequality.
$(\frac{2}{3},\frac{1}{8})$

• A. Yes
• B. No
• C. Cannot be determined

Answer: (B)

Graphing Linear Inequalities in Two Variable

Graphing two-dimensional inequalities involves a few straightforward steps that build on the concepts of one-dimensional inequalities. When dealing with inequalities like \( x \geq 1 \), the first step is to graph the corresponding line, which in this case is a vertical line at \( x = 1 \). This line divides the plane into two regions. To determine which side to shade, you need to identify the points that satisfy the inequality. For \( x \geq 1 \), you would shade the region to the right of the line, as all points in that area have x-coordinates greater than or equal to 1.

When graphing inequalities involving both x and y, such as \( y < x \), you start by graphing the line \( y = x \). Since the inequality is strict (less than), you would use a dashed line to indicate that points on the line are not included in the solution set. The next step is to determine which side of the line to shade. This can be done by testing a point not on the line, such as \( (0, 0) \). If you substitute these coordinates into the inequality and find a true statement, you shade the side of the line that includes that point. If the statement is false, you shade the opposite side.

For more complex inequalities like \( y > 2x - 4 \), the process is similar. First, graph the line \( y = 2x - 4 \) using a dashed line since the inequality is strict. Next, test a point, such as \( (0, 1) \). Substituting these values into the inequality shows that \( 1 > 2(0) - 4 \) simplifies to \( 1 > -4 \), which is true. Therefore, you would shade the region above the line, indicating that all points in that area satisfy the inequality.

In summary, when graphing linear inequalities, always start by graphing the corresponding line, determine whether to use a solid or dashed line based on the inequality symbol, and then test a point to decide which side of the line to shade. If the inequality can be rearranged into slope-intercept form \( y = mx + b \), you can quickly identify that for \( y > mx + b \), you shade above the line, and for \( y < mx + b \), you shade below the line. This method provides a clear visual representation of the solution set for the inequality.

Graphing Linear Inequalities in Two Variable Example 1

In this exploration of systems of inequalities, we analyze three equations and their corresponding graphs. The first equation, \( y > -3x + 4 \), represents a line with a y-intercept of 4 and a slope of -3. To visualize this, we consider the line as if it were an equation of equality, which helps us identify the graph. Since the inequality is a 'greater than' symbol, the line is dashed, indicating that points on the line are not included in the solution set. The area above the line is shaded, representing all the solutions where y is greater than the line.

Next, we examine the second equation, \( x + y < 1 \). Rewriting this in slope-intercept form gives us \( y < -x + 1 \). This line has a y-intercept of 1 and a slope of -1. Similar to the first equation, the 'less than' symbol indicates a dashed line, and we shade below the line to represent the solutions where y is less than the line.

Finally, we look at the third equation, \( -3x - y \geq -4 \). By flipping the signs, we can rewrite it as \( 3x + y \leq 4 \). This transformation also flips the inequality symbol. In slope-intercept form, this becomes \( y \leq -3x + 4 \). Here, the y-intercept remains 4, and the slope is still -3. However, since we have a 'less than or equal to' symbol, the line is solid, indicating that points on the line are included in the solution set, and we shade below the line.

In summary, we matched each equation to its graph based on the characteristics of the lines and the direction of shading determined by the inequality symbols. Understanding how to manipulate equations into slope-intercept form and recognizing the implications of different inequality symbols are crucial skills in graphing systems of inequalities.

Graph the inequality

• A.
• B.
• C.
• D.

Answer: (C)

Here’s what students ask on this topic:

How do you graph a linear inequality in two variables?

To graph a linear inequality in two variables, first graph the corresponding line by replacing the inequality symbol with an equal sign. Use a solid line if the inequality includes ≤ or ≥ and a dashed line if it is < or >. Next, select a test point not on the line, often (0,1) or (0,0), and substitute its coordinates into the inequality. If the inequality holds true, shade the region containing that point; otherwise, shade the opposite side. This shaded region represents all solutions to the inequality, unlike equations where solutions lie only on the line.

What is the difference between graphing linear equations and linear inequalities in two variables?

Graphing linear equations in two variables involves plotting a line where all points satisfy the equation exactly. For linear inequalities, you graph the boundary line (solid for ≤ or ≥, dashed for < or >) and then shade the region where the inequality holds true. This shaded area represents all solutions, not just points on the line. Thus, inequalities have solution regions, while equations have solution lines.

How do you determine which side of the line to shade when graphing a linear inequality?

After graphing the boundary line of the inequality, pick a test point not on the line, such as (0,0) if it is not on the line. Substitute the test point's coordinates into the inequality. If the inequality is true, shade the side of the line containing that point. If false, shade the opposite side. This method ensures you shade the region where the inequality holds true.

When should you use a solid line versus a dashed line in graphing linear inequalities?

Use a solid line when the inequality includes equality, such as ≤ (less than or equal to) or ≥ (greater than or equal to). This means points on the line satisfy the inequality. Use a dashed line when the inequality is strict, such as < (less than) or > (greater than), indicating points on the line are not included in the solution set.

Can you explain how to graph the inequality $y>2x-4$?

First, graph the boundary line $y=2x-4$. Since the inequality is > (strict), draw this line as dashed. Next, choose a test point not on the line, such as (0,0). Substitute into the inequality: $0\; >\; 2(0)\; -\; 4$ simplifies to $0\; >\; -4$, which is true. Therefore, shade the region above the line, representing all points where $y$ is greater than $2x\; -\; 4$.

What does the solution set of a linear inequality in two variables represent?

The solution set of a linear inequality in two variables represents all ordered pairs (x, y) that satisfy the inequality. Graphically, this is shown as a shaded region on one side of the boundary line. Unlike equations where solutions lie exactly on the line, inequalities include all points in the shaded area, indicating a range of possible values for x and y.