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# Linear Inequalities in Two Variables

## Graph inequalities like 2x + 3 < y on the coordinate plane

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Graphs of Linear Inequalities.

### Vocabulary

Inequality
An inequality is a statement that shows a relationship between two expressions that are not always equal. An inequality is written using one of the following inequality symbols: greater than \begin{align*}(>)\end{align*}; less than \begin{align*}(<)\end{align*}; greater than or equal to \begin{align*}(\ge)\end{align*}; less than or equal to \begin{align*}(\le)\end{align*}. The solution to an inequality is indicated by a shaded region that contains all the ordered pairs that satisfy the inequality.

A linear equation is a line when graphed. What about a linear inequality? Can you represent the following linear inequality in the Cartesian plane?

\begin{align*}2x-3y < 6\end{align*}

### Guidance

When a linear inequality is graphed on a grid, the solution will appear as a shaded area, as opposed to simply a straight line. The general form of a linear inequality with two variables is \begin{align*}ax+by>c\end{align*} or \begin{align*}ax+by, where ‘\begin{align*}a\end{align*}’ and ‘\begin{align*}b\end{align*}’ are the coefficients of the variables and are not both equal to zero. The constant is ‘\begin{align*}c\end{align*}’.

The inequality symbol \begin{align*}(>)\end{align*} is read “greater than” and the inequality symbol \begin{align*}(<)\end{align*} is read “less than”. The inequality symbol \begin{align*}(\ge)\end{align*} is read “greater than or equal to” and the inequality symbol \begin{align*}(\le)\end{align*} is read “less than or equal to”. The inequality symbol determines whether the line is dashed or solid.

• All inequalities that have the symbol \begin{align*}(>)\end{align*} or \begin{align*}(<)\end{align*} are graphed with a dashed line.
• All inequalities that have the symbol \begin{align*}(\ge)\end{align*} or \begin{align*}(\le)\end{align*} are graphed with a solid line.

Don't forget that when an inequality is divided or multiplied by a negative number, the direction of the inequality sign is reversed.

#### Example A

Graph the inequality \begin{align*}3x+4y \le 12\end{align*}.

Solution:

\begin{align*}& 3x+4y \le 12 && \text{Write the inequality in slope intercept form } (y=mx+b).\\ & 3x{\color{red}-3x}+4y \le {\color{red}-3x}+12\\ & 4y \le {\color{red}-3x}+12\\ & \frac{4y}{{\color{red}4}} \le \frac{-3x}{{\color{red}4}}+\frac{12}{{\color{red}4}}\\ & \frac{\cancel{4}y}{\cancel{4}} \le \frac{-3x}{4}+\frac{\overset{{\color{red}3}}{\cancel{12}}}{\cancel{4}}\\ & \boxed{y \le -\frac{3}{4}x+3}\end{align*}

The above graph represents the graph of the inequality before the solution set region is shaded. The line is a solid line because the inequality symbol is \begin{align*}(\le)\end{align*}. To determine whether to shade above the line or below the line, choose a point that is not on the line and test its coordinates in the original inequality. If the coordinates of the point satisfy the inequality, then the area above or below the line, containing the point, will be shaded. If the coordinates of the point do not satisfy the inequality, then the area above or below the line, that does not contain the test point, will be shaded. The point (1, 1) is not on the graphed line. The point will be tested to determine if the coordinates satisfy the inequality.

\begin{align*}3x+4y &\le 12\\ 3({\color{red}1})+4({\color{red}1}) &\le 12 && \text{Substitute} \ (1, 1) \ \text{for} \ x \ \text{and} \ y \ \text{of the original inequality.}\\ {\color{red}3}+{\color{red}4} &\le 12 && \text{Evaluate the inequality.}\\ {\color{red}7} &\le 12 && \text{Is it true?}\end{align*}

Yes, seven is less than 12. The point (1, 1) satisfies the inequality. Therefore, the solution set is all of the area below the line that contains the point (1, 1). The ordered pair that satisfies the inequality will lie within the shaded region.

The solution set for the inequality is the entire shaded region shown in the graph. The solid line means that all of the points on the line will satisfy the inequality. In general, you will shade below the line if the inequality is of the form \begin{align*}y or \begin{align*}y\le mx+b\end{align*}.

#### Example B

Graph the inequality \begin{align*}2x-4y < -12\end{align*}.

Solution:

\begin{align*}& 2x-4y < -12 && \text{Write the inequality in slope intercept form} \ (y=mx+b).\\ & 2x{\color{red}-2x}-4y < {\color{red}-2x}-12\\ & -4y < {\color{red}-2x}-12\\ & \frac{-4y}{{\color{red}-4}} < \frac{-2x}{{\color{red}-4}} -\frac{12}{{\color{red}-4}}\\ & \frac{\cancel{-4}y}{\cancel{-4}}<\frac{-2x}{-4}-\frac{\overset{{\color{red}-3}}{\cancel{12}}}{\cancel{-4}}\\ & \boxed{y \ {\color{red}>} \ \frac{1}{2}x+3}\end{align*}

Note that the inequality was divided by negative 4 which caused the inequality sign to reverse its direction.

The above graph represents the graph of the inequality before the solution set region is shaded. The line is a dashed line because the inequality symbol is \begin{align*}(<)\end{align*}. To determine whether to shade above the line or below the line, choose a point that is not on the line and test its coordinates in the original inequality. If the coordinates of the point satisfy the inequality, then the area above or below the line, containing the point, will be shaded. If the coordinates of the point do not satisfy the inequality, then the area above or below the line, that does not contain the test point, will be shaded. The point (1, 1) is not on the graphed line. The point will be tested to determine if the coordinates satisfy the inequality.

\begin{align*}2x-4y &< -12\\ 2({\color{red}1})-4({\color{red}1}) &< -12 && \text{Substitute} \ (1, 1) \ \text{for} \ x \ \text{and} \ y \ \text{of the original inequality.}\\ {\color{red}2-4} &< -12 && \text{Evaluate the inequality.}\\ {\color{red}-2} &< -12 && \text{Is it true?}\end{align*}

No, negative two is greater than negative twelve. The point (1, 1) does not satisfy the inequality. Therefore, the solution set is all of the area above the line that does not contain the point (1, 1). The ordered pair does not satisfy the inequality and will not lie within the shaded region.

The solution set for the inequality is the entire shaded region shown in the graph. The dashed or dotted line means that none of the points on the line will satisfy the inequality. In general, you will shade above the line if the inequality is of the form \begin{align*}y>mx+b\end{align*} or \begin{align*}y\ge mx+b\end{align*}.

#### Example C

For the following, determine the inequality, in slope-intercept form, that is graphed.

Solution: Begin by determining the slope of the line. The slope of the line is determined by counting 2 units to the right and 3 units upward. The slope of the line for this graph is

\begin{align*}m=\frac{rise}{run}={\color{red}\frac{3}{2}}\end{align*}

The \begin{align*}y\end{align*}-intercept for the line is (0, –2). The equation of the line in slope-intercept form is

\begin{align*}y=\frac{3}{2}x-2\end{align*}

The solution set is found in the shaded region that is above the line. The line is a solid line. Therefore the inequality symbol that must be inserted is greater than or equal to. The inequality that is modeled by the above graph is:

\begin{align*}\boxed{y \ge \frac{3}{2}x-2}\end{align*}

#### Concept Problem Revisited

\begin{align*}2x-3y < 6\end{align*}

The first step is to rearrange the inequality in slope-intercept form. This process is the same as it is for linear equations.

\begin{align*}& 2x-3y < 6\\ & 2x{\color{red}-2x}-3y < {\color{red}-2x}+6\\ & -3y < -2x+6\\ & \frac{-3y}{{\color{red}-3}} < \frac{-2x}{{\color{red}-3}} + \frac{6}{{\color{red}-3}}\\ & \frac{\cancel{-3}y}{\cancel{-3}} < \frac{-2x}{-3} + \frac{\overset{{\color{red}-2}}{\cancel{6}}}{\cancel{-3}}\\ & \boxed{y \ {\color{red}>} \ \frac{2}{3}x-2}\end{align*}

Note that the inequality was divided by negative 3 which caused the inequality sign to reverse its direction.

The graph of the inequality is done the same as it is for a linear equation. In this case the graph will be a dashed or dotted line because the sign is greater than \begin{align*}(>)\end{align*}.

The above graph represents the graph of the inequality before the solution set region is shaded. To determine whether to shade above the line or below the line, choose a point that is not on the line and test its coordinates in the original inequality. If the coordinates of the point satisfy the inequality, then the area above or below the line, containing the point, will be shaded. If the coordinates of the point do not satisfy the inequality, then the area above or below the line, that does not contain the test point, will be shaded.

The point (1, 1) is not on the graphed line. The point will be tested to determine if the coordinates satisfy the inequality.

\begin{align*}2x-3y &< 6\\ 2({\color{red}1})-3({\color{red}1}) &< 6 && \text{Substitute} \ (1, 1) \ \text{for} \ x \ \text{and} \ y \ \text{of the original inequality.}\\ {\color{red}2}- {\color{red}3} &< 6 && \text{Evaluate the inequality.}\\ {\color{red}-1} &< 6 && \text{Is it true?}\end{align*}

Yes, negative one is less than six. The point (1, 1) satisfies the inequality. Therefore, the solution set is all of the area above the line that contains the point (1, 1). The ordered pair that satisfies the inequality will lie within the shaded region.

The solution set for the inequality is the entire shaded region shown in the graph. The dashed or dotted line means that none of the points on the line will satisfy the inequality.

### Guided Practice

1. Without graphing, determine if each point is in the shaded region for each inequality.

i) (2, –3) and \begin{align*}2y<-3x+1\end{align*}
ii) (–3, 5) and \begin{align*}-3x > 2y+6\end{align*}

2. Graph the following inequality: \begin{align*}5x-3y \ge 15\end{align*}

3. Determine the inequality that models the following graph:

1. If the point satisfies the inequality, then the point will lie within the shaded region. Substitute the coordinates of the point into the inequality and evaluate the inequality. If the solution is true, then the point is in the shaded area.

i)
\begin{align*}2y &< -3x+1 && \text{Substitute} \ (2, -3) \ \text{for} \ x \ \text{and} \ y \ \text{in the inequality.}\\ 2({\color{red}-3}) &< -3({\color{red}2})+1 && \text{Evaluate the inequality.}\\ {\color{red}-6} &< {\color{red}-6}+1\\ -6 &< {\color{red}-5} && \text{Is it true?}\end{align*}
Yes, negative six is less than negative five. The point (2, –3) satisfies the inequality. The ordered pair will lie within the shaded region.
ii)
\begin{align*}-3x &> 2y+6 && \text{Substitute} \ (-3, 5) \ \text{for} \ x \ \text{and} \ y \ \text{in the inequality.}\\ -3({\color{red}-3}) &> 2({\color{red}5})+6 && \text{Evaluate the inequality.}\\ {\color{red}9} &> {\color{red}10}+6\\ 9 &> {\color{red}16} && \text{Is it true?}\end{align*}
No, nine is not greater than sixteen. The point (–3, 5) does not satisfy the inequality. The ordered pair does not satisfy the inequality and will not lie within the shaded region.

2.

\begin{align*}& 5x-3y \ge 15 && \text{Write the inequality in slope intercept form} \ (y=mx+b).\\ & 5x{\color{red}-5x}-3y \ge {\color{red}-5x}+15\\ & -3y \ge -5x+15\\ & \frac{-3y}{{\color{red}-3}} \ge \frac{-5x}{{\color{red}-3}} + \frac{15}{{\color{red}-3}}\\ & \frac{\cancel{-3}y}{\cancel{-3}} \ge \frac{5}{3}x + \frac{\overset{{\color{red}-5}}{\cancel{15}}}{\cancel{-3}}\\ & \boxed{y \ {\color{red}\le} \ \frac{5}{3}x-5}\end{align*}
Note that the inequality was divided by negative 3 which caused the inequality sign to reverse its direction.
Is the graph of the inequality shaded correctly?
The point (1, 1) is not on the graphed line. The point will be tested to determine if the coordinates satisfy the inequality.
\begin{align*}5x-3y &\ge 15\\ 5({\color{red}1})-3({\color{red}1}) &\ge 15 && \text{Substitute} \ (1, 1) \ \text{for} \ x \ \text{and} \ y \ \text{of the original inequality.}\\ {\color{red}5}{\color{red}-3} &\ge 15 && \text{Evaluate the inequality.}\\ {\color{red}2} &\ge 15 && \text{Is it true?}\end{align*}
No, two is not greater than or equal to fifteen. The point (1, 1) does not satisfy the inequality. The ordered pair does not satisfy the inequality and will not lie within the shaded region.

3. Begin by determining the slope of the line. The slope of the line is determined by counting 2 units to the left and 4 units downward. The slope of the line for this graph is

\begin{align*}m=\frac{rise}{run}={\color{red}\frac{-4}{-2}}={\color{red}2}\end{align*}
The \begin{align*}y\end{align*}-intercept for the line is (0, –3). The equation of the line in slope-intercept form is \begin{align*}y=2x-3\end{align*}
The solution set is found in the shaded region that is below the line. The line is a solid line. Therefore the inequality symbol that must be inserted is less than or equal to. The inequality that is modeled by the above graph is:
\begin{align*}\boxed{y \le 2x-3}\end{align*}

### Practice

Without graphing, determine if each point is in the shaded region for each inequality.

1. (2, 1) and \begin{align*}2x+y>5\end{align*}
2. (–1, 3) and \begin{align*}2x-4y \le -10\end{align*}
3. (–5, –1) and \begin{align*}y > -2x+8\end{align*}
4. (6, 2) and \begin{align*}2x+3y \ge -2\end{align*}
5. (5, –6) and \begin{align*}2y< 3x+3\end{align*}

Determine the inequality that is modeled by each of the following graphs.

1. .

1. .

1. .

1. .

1. .

Graph the following linear inequalities on a Cartesian plane.

1. \begin{align*}4x-2y>8\end{align*}
2. \begin{align*}4y-3x \le -8\end{align*}
3. \begin{align*}x-y < -3\end{align*}
4. \begin{align*}3x+y > -1\end{align*}
5. \begin{align*}x+3y \ge 9\end{align*}

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