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 like (0,1) helps determine which side to shade. For example, the inequality $y>2x-4$ is graphed by plotting the line $y=2x-4$ and shading above it. Solutions to inequalities form regions, unlike equations where solutions lie on lines, emphasizing the importance of understanding multivariable polynomials and their graphical representations.

Intro to Linear Inequalities in Two Variables

Understanding solutions to two-variable linear inequalities builds on the concept of ordered pairs that satisfy two-variable equations. An ordered pair (x, y) is a solution to an equation if substituting 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 point lies on the line represented by the equation.

When dealing with linear inequalities in two variables, the approach is similar but with key differences. 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, substitute the values of x and y into the inequality. If the inequality holds true, the point is a solution; if not, it is not.

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 shaded regions rather than just a line. The line itself corresponds to the boundary where ax + by = c. Points that satisfy the inequality lie within the shaded region either above or below this boundary, depending on the inequality symbol.

Testing another point, such as (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 important to note the role of 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 is inclusive (e.g., ≤ or ≥), points on the line satisfy the inequality and are part of the solution set.

In summary, solving linear inequalities in two variables involves substituting ordered pairs into the inequality to verify if they satisfy the condition, understanding that solutions form a region on the graph rather than just a line, and recognizing the significance of the inequality symbol in determining whether the boundary line is included in the solution set.

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. For example, to graph $y>2x-4$, graph the line $y=2x-4$ with a dashed line and shade above it.

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

The key difference is that linear equations in two variables are graphed as lines, where solutions lie exactly on the line. In contrast, linear inequalities represent regions on the graph. When graphing an inequality, you first graph the boundary line (solid for ≤ or ≥, dashed for < or >). Then, you shade the region where the inequality holds true, which includes infinitely many points either above, below, or to one side of the line. For example, the equation $y=x$ is a line, but the inequality $y<x$ includes all points below the line, forming a shaded region.

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

To determine which side of the line to shade, pick a test point not on the boundary line, commonly (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. For example, for the inequality $y>2x-4$, testing (0,0) gives $0>2*0-4$ or $0\; >\; -4$, which is true, so shade the region above the line.

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

Use a solid line when the inequality includes equality, that is, ≤ (less than or equal to) or ≥ (greater than or equal to). This indicates points on the line satisfy the inequality. Use a dashed line when the inequality is strict, < (less than) or > (greater than), meaning points on the line do not satisfy the inequality. For example, graphing $y\le x$ uses a solid line, while $y<x$ uses a dashed line.

How can you verify if a point is a solution to a linear inequality in two variables?

To verify if a point is a solution, substitute its coordinates into the inequality. If the resulting statement is true, the point is a solution; if false, it is not. For example, for the inequality , test the point (-1, 0): $2*(-1)\; +\; 3*0\; =\; -2$, and since is true, (-1, 0) is a solution. Testing (5, 3) gives $2*5\; +\; 3*3\; =\; 19$, and is false, so (5, 3) is not a solution.