2.10: Testing Solutions for Linear Inequalities in Two Variables
A taxi cab charges $2 per mile plus $0.20 per stopped minute in traffic. If your cab bill totals less than $10 but more than $5, which of the following could have occurred during your ride?
A. You traveled 5 miles and sat in traffic for 3 minutes. B. You traveled 2 miles and sat in traffic for 2 minutes. C. You traveled 4 miles and sat in traffice for 6 minutes.
Watch This
James Sousa: Ex: Determine if Ordered Pairs Satisfy a Linear Inequality
Guidance
A linear inequality is very similar to the equation of a line, but with an inequality sign. They can be written in one of the following ways:
\begin{align*}Ax + By < C && Ax + By > C && Ax + By \le C && Ax + By \ge C\end{align*}
Notice that these inequalities are very similar to the standard form of a line. We can also write a linear inequality in slope-intercept form.
\begin{align*}y < mx + b && y > mx + b && y \le mx + b && y \ge mx + b\end{align*}
In all of these general forms, the \begin{align*}A, B, C, m\end{align*}, and \begin{align*}b\end{align*} represent the exact same thing they did with lines.
An ordered pair, or point, is a solution to a linear inequality if it makes the inequality true when the values are substituted in for \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.
Example A
Which ordered pair is a solution to \begin{align*}4x - y > -12\end{align*}?
a) (6, -5)
b) (-3, 0)
c) (-5, 4)
Solution: Plug in each point to see if they make the inequality true.
a) \begin{align*}4(6) - (-5) & >-12\\ 24 + 5 & > -12\\ 29 & > -12\end{align*}
b)\begin{align*}4(-3)-0 & >-12\\ -12 & \ \cancel{>} - 12\end{align*}
c)\begin{align*}4(-5) - 4 & > -12\\ -20 -4 & > -12\\ -24 & \ \cancel{>} -12\end{align*}
Of the three points, a) is the only one where the inequality holds true. b) is not true because the inequality sign is only “greater than,” not “greater than or equal to.”
Example B
Is the point (-9, 1) a solution for \begin{align*} y < 5x + 1 \end{align*}
Solution: Substitute in the point values for x and y and see if the inequality holds true.
\begin{align*}1<5 \cdot -9 +1 \\ 1<-45+1 \\ 1<-44 \end{align*}
This is false. Therefore, (-9, 1) is not a solution.
Example C
Determine 3 solutions to the inequality \begin{align*}2x-7y > -12\end{align*}
Solution: Select values for x and y that would make the inequality true. If \begin{align*}x=2\end{align*} and \begin{align*}y=-2\end{align*}, the inequality is true, \begin{align*}4+14 > -12\end{align*}. Another easy point would be the origin. Testing it, we have \begin{align*}0>-12\end{align*}. Lastly, we could select a point where y is zero and the x value is postive. For exmaple, the points (1, 0), (2, 0), (3, 0), etc... would all work. There are infinitely many solutions.
Intro Problem Revisit To solve the taxi cab problem, we must first set up an inequality to represent the scenario.
\begin{align*} 5 < 2x + 0.2y < 10\end{align*}, where x equals the miles traveled and y equals the number of minutes in stopped traffic.
Now let's test each of the possibilities to see if they fit the inequality.
A: \begin{align*}2(5) + 0.2(3) = 10 + 0.6 = 10.6 > 10\end{align*} so this possibility could not have occurred. B: \begin{align*}2(2) + 0.2(2) = 4 + 0.4 = 4.4 <5\end{align*} so this possibility could not have occurred. C: \begin{align*}2(4) + 0.2(6) = 8 + 1.2 = 9.2\end{align*}; \begin{align*}5 < 9.2 < 10\end{align*} so this possibility could have occurred.
Vocabulary
- Linear Inequality
- An inequality, usually in two variables, of the form \begin{align*}Ax +By < C, Ax + By > C, Ax +By \le C\end{align*}, or \begin{align*}Ax + By \ge C\end{align*}.
- Solution
- An ordered pair that satisfies a given inequality.
Guided Practice
1. Which inequality is (-7, 1) a solution for?
a) \begin{align*}y < 2x - 1\end{align*}
b) \begin{align*}4x -3y \ge 9\end{align*}
c) \begin{align*}y > -4\end{align*}
2. List three possible solutions for \begin{align*}5x - y \le 3\end{align*}.
Answers
1. Plug (-7, 1) in to each equation. With c), only use the \begin{align*}y-\end{align*}value.
a) \begin{align*}1 &< 2(-7) -1\\ 1 & \ \bcancel{<} - 15\end{align*}
b) \begin{align*}4(-7) -3(1) &\ge 9\\ -28 -3 &\ge 9\\ -31 & \ \cancel{\ge} \ 9\end{align*}
c) \begin{align*}1 > -4\end{align*}
(-7, 1) is only a solution to \begin{align*}y > -4\end{align*}.
2. To find possible solutions, plug in values to the inequality. There are infinitely many solutions. Here are three: (-1, 0), (-4, 3), and (1, 6).
\begin{align*}5(-1) -0 & \le 3 && 5(-4) -3 \le 3 && 5(1) -6 \le 3\\ -5 & \le 3 && \qquad -17 \le 3 && \quad \ \ -1 \le 3\end{align*}
Practice
Using the four inequalities below, determine which point is a solution for each one. There may be more than one correct answer. If the answer is none, write none of these.
A) \begin{align*}y \le \frac{2}{3}x - 5\end{align*}
B) \begin{align*}5x +4y > 20\end{align*}
C) \begin{align*}x - y \ge -5\end{align*}
D) \begin{align*}y > -4x + 1\end{align*}
- (9, -1)
- (0, 0)
- (-1, 6)
- (-3, -10)
Determine which inequality each point is a solution for. There may be more than one correct answer. If the answer is none, write none of these.
A) (-5, 1)
B) (4, 2)
C) (-12, -7)
D) (8, -9)
- \begin{align*}2x -3y > 8\end{align*}
- \begin{align*}y \le -x -4\end{align*}
- \begin{align*}y \ge 6x + 7\end{align*}
- \begin{align*}8x +3y < -3\end{align*}
- Is (-6, -8) a solution to \begin{align*}y < \frac{1}{2}x -6\end{align*}?
- Is (10, 1) a solution to \begin{align*}y \ge -7x + 1\end{align*}?
For problems 11-15, find three solutions for each inequality.
- \begin{align*}5x -y >12\end{align*}
- \begin{align*}y \le -2x + 9\end{align*}
- \begin{align*}y \ge -4\end{align*}
- \begin{align*}3x + 4y < -5\end{align*}
- \begin{align*}x \le 7\end{align*}
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Linear Inequality
Linear inequalities are inequalities that can be written in one of the following four forms: , or .solution
A solution to an equation or inequality should result in a true statement when substituted for the variable in the equation or inequality.Image Attributions
Here you'll learn how to determine if an ordered pair is a solution to a linear inequality in two variables.