Testing Solutions for Linear Inequalities in Two Variables

Substitute given values for variables in both equations and compare the results

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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 travelled 5 miles and sat in traffic for 3 minutes.

B. You travelled 2 miles and sat in traffic for 2 minutes.

C. You travelled 4 miles and sat in traffic for 6 minutes.

Solution for Linear Inequality

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*}.

Let's determine the solutions to the following inequalities.

1. Which ordered pair is a solution to \begin{align*}4x - y > -12\end{align*}?

Plug in each point to see if they make the inequality true.

a) (6, -5)

\begin{align*}4(6) - (-5) & >-12\\ 24 + 5 & > -12\\ 29 & > -12\end{align*}

b) (-3, 0)

\begin{align*}4(-3)-0 & >-12\\ -12 & \ \cancel{>} - 12\end{align*}

c) (-5, 4)

\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.”

1. Is the point (-9, 1) a solution for \begin{align*} y < 5x + 1 \end{align*}?

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.

1. Determine 3 solutions to the inequality \begin{align*}2x-7y > -12\end{align*}.

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 positive. For example, the points (1, 0), (2, 0), (3, 0), etc... would all work. There are infinitely many solutions.

Examples

Example 1

Earlier, you were asked to find which scenario could have occurred during your taxi ride.

To solve this 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.

Example 2

Which inequality is (-7, 1) a solution for?

Plug (-7, 1) in to each equation.

1. \begin{align*}y < 2x - 1\end{align*}
\begin{align*}1 &< 2(-7) -1\\ 1 & \ \bcancel{<} - 15\end{align*}
1. \begin{align*}4x -3y \ge 9\end{align*}
\begin{align*}4(-7) -3(1) &\ge 9\\ -28 -3 &\ge 9\\ -31 & \ \cancel{\ge} \ 9\end{align*}
1. \begin{align*}y > -4\end{align*}

\begin{align*}1 > -4\end{align*}

(-7, 1) is only a solution to #3, \begin{align*}y > -4\end{align*}.

Example 3

List three possible solutions for \begin{align*}5x - y \le 3\end{align*}.

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*}

Review

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*}

1. (9, -1)
2. (0, 0)
3. (-1, 6)
4. (-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)

1. \begin{align*}2x -3y > 8\end{align*}
2. \begin{align*}y \le -x -4\end{align*}
3. \begin{align*}y \ge 6x + 7\end{align*}
4. \begin{align*}8x +3y < -3\end{align*}
5. Is (-6, -8) a solution to \begin{align*}y < \frac{1}{2}x -6\end{align*}?
6. Is (10, 1) a solution to \begin{align*}y \ge -7x + 1\end{align*}?

For problems 11-15, find three solutions for each inequality.

1. \begin{align*}5x -y >12\end{align*}
2. \begin{align*}y \le -2x + 9\end{align*}
3. \begin{align*}y \ge -4\end{align*}
4. \begin{align*}3x + 4y < -5\end{align*}
5. \begin{align*}x \le 7\end{align*}

To see the Review answers, open this PDF file and look for section 2.10.

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Color Highlighted Text Notes

Vocabulary Language: English

TermDefinition
Linear Inequality Linear inequalities are inequalities that can be written in one of the following four forms: $ax + b > c, ax + b < c, ax + b \ge c$, or $ax + b \le c$.
solution A solution to an equation or inequality should result in a true statement when substituted for the variable in the equation or inequality.