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# 8.3: Testing for Truth

Difficulty Level: At Grade Created by: CK-12

This activity is intended to supplement Algebra I, Chapter 7, Lesson 6.

## Problem 1 - Is a Point a Solution?

In this activity, you will be determining if random points are solutions to inequalities. Before beginning the activity, you will need to set up the random number generator. Change the random seed using the last 4 digits of your phone number. Enter the digits, then rand on the home screen and press ENTER.

Below, you are given the inequality y>x2\begin{align*}y > -x - 2\end{align*} and the coordinates of a point. Next, create 2 lists of 3 random numbers. On the home screen, enter randInt(-5,10,3)ALPHA[L1]\begin{align*}[L_1]\end{align*}. Repeat, replacing L1\begin{align*}L_1\end{align*} with L2\begin{align*}L_2\end{align*} to generate the y\begin{align*}y-\end{align*}list. Press STAT and select 1:Edit... to see the x\begin{align*}x-\end{align*} and y\begin{align*}y-\end{align*}values in L1\begin{align*}L_1\end{align*} and L2\begin{align*}L_2\end{align*}. Using the table below, determine whether or not the point is a solution of the inequality.

Point A (x,y)\begin{align*}A \ (x,y)\end{align*} y\begin{align*}y\end{align*} x2\begin{align*}-x -2\end{align*} y>x2\begin{align*}y > -x -2\end{align*} T or F
(2,2)\begin{align*}(2,2)\end{align*} 2\begin{align*}2\end{align*} 22\begin{align*}-2 -2\end{align*} 2>4\begin{align*}2 > -4\end{align*} T

You can check each equation on the home screen using the : menu. You can also check the solution by graphing the inequality, tracing to the given x\begin{align*}x-\end{align*}value and checking to see if the y\begin{align*}y-\end{align*}value corresponds.

## Problem 2 - Generating Solutions

In this problem, you are given two inequalities, y4\begin{align*}y \le 4\end{align*} and y>2\begin{align*}y > -2\end{align*}. Again, generate random numbers in L1\begin{align*}L_1\end{align*} (or x\begin{align*}x\end{align*}) and L2\begin{align*}L_2\end{align*} (or y\begin{align*}y\end{align*}). True is 1\begin{align*}1\end{align*} and False is 0\begin{align*}0\end{align*}.

Complete the table below. Generate coordinates until you find at least one solution to the inequality.

Point (x,y)\begin{align*}(x,y)\end{align*} Test: y4\begin{align*}y \le 4\end{align*} (T or F) Test: y>2\begin{align*}y > -2\end{align*} (T or F) Final answer? (T or F)
ex: (2,0)\begin{align*}(2,0)\end{align*}

04\begin{align*}0 \le 4\end{align*}

T

02\begin{align*}0 \ge -2\end{align*}

T

T

## Problem 3 - Overlapping Regions

In this problem, three inequalities intersect to form a triangular region. Again, generate random numbers in L1\begin{align*}L_1\end{align*} and L2\begin{align*}L_2\end{align*} (for X\begin{align*}X\end{align*} and Y\begin{align*}Y\end{align*}), and see if the coordinates are solutions to the system. Complete the table below, making sure you have found at least one solution.

You can test the equations on the home screen using the list of elements instead of each individual coordinate.

You can graph each inequality and test a point by moving a free-floating cursor to approximately each coordinate in the table to see if it falls inside the shaded region.

Point (x,y)\begin{align*}(x,y)\end{align*} Test: y0.25x+4\begin{align*}y \le 0.25x + 4\end{align*} (T or F) Test: y2x1\begin{align*}y \ge -2x - 1\end{align*} (T or F) Test: yx+2\begin{align*}y \ge x + 2\end{align*} (T or F) Final answer? (T or F)
ex: (2,0)\begin{align*}(2,0)\end{align*}

00.25(2)+4\begin{align*}0 \le 0.25(2) + 4\end{align*}

00.5+4\begin{align*}0 \le 0.5 + 4\end{align*}

04.5\begin{align*}0 \le 4.5\end{align*}

T

02(2)1\begin{align*}0 \ge -2(2) - 1\end{align*}

041\begin{align*}0 \ge - 4 - 1\end{align*}

05\begin{align*}0 \ge -5\end{align*}

T

02+2\begin{align*}0 \ge 2 + 2\end{align*}

04\begin{align*}0 \ge 4\end{align*}

F

F

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