Problem 46 - Algebra 2

Question

For Exercises \(46-49,\) use the following information. You can use the operators in the LOGIC menu on the TI-83/84 Plus to graph compound and absolute value inequalities. To display the LOGIC menu, press 2nd Test. Clear the \(Y=\) list. Enter \((5 x+2 > 12)\) and \((3 x-8 < 1)\) as \(Y 1\) . With your calculator in DOT mode and using the standard viewing window, press GRAPH. Make a sketch of the graph displayed.

Step-by-Step Solution

Answer

Graph the inequalities (5x+2 > 12) and (3x-8 < 1) on a calculator, using DOT mode and the standard window to visualize their solution regions.

Step 1: Access the Logic Menu on Calculator

Turn on your TI-83/84 Plus calculator and align it to the home screen. Press the "2nd" button and then the "TEST" button. This will navigate you to the LOGIC menu required for entering inequalities.

Step 2: Clear Previous Graph Entries

Press the "Y=" button to view any previous graph entries. Use the "CLEAR" button to remove any existing equations from the list. This ensures that only the current inequalities are graphed.

Step 3: Enter the First Inequality

In the Y1 entry space, enter the inequality (5x + 2 > 12). Use the inequality operator menu accessed previously for the '>' symbol. You should enter 5x + 2 (>, by using the LOGIC menu) 12.

Step 4: Enter the Second Inequality

In the Y2 entry space, enter the inequality (3x - 8 < 1). Again, use the inequality operator from the LOGIC menu for the '<' symbol. Ensure the expression is entered as 3x - 8 (<, by using the LOGIC menu) 1.

Step 5: Set Calculator to DOT Mode

Press the "MODE" button and scroll down to find the option for DOT mode. Use the arrow keys to select it. This will display the graph with dots and helps in distinguishing multiple graphs on the same viewing window.

Step 6: Use Standard Viewing Window

Press the "ZOOM" button and then select '6' which is the standard viewing window. This sets the window to X[-10,10] and Y[-10,10] with standard scaling, suitable for viewing both graph inequalities concurrently.

Step 7: Display the Graph

Press the "GRAPH" button. The calculator will plot both inequalities' shaded regions. The regions highlight where each inequality holds true.

Step 8: Sketch the Graph Displayed

Observe the sections where the shaded regions overlap and where they do not overlap. Sketch the graph with these shaded regions as they are visible, using dot-shading to match the calculator's display method.

Key Concepts

Compound Inequalities

Compound inequalities combine two separate inequalities, offering a visual intersection or union of solutions. They can be connected using 'and' or 'or'. In mathematical terms, an 'and' compound inequality means that both conditions must be true simultaneously, while an 'or' compound inequality requires only one to be true.

For example, let's look at the compound inequality - \(5x + 2 > 12\)- \(3x - 8 < 1\)This is an 'and' scenario, where we seek the region where both conditions hold true. Graphically, the overlap of these inequalities is shaded, indicating the solution set. These inequalities are usually solved graphically using graphing calculators, like the TI-83/84, which utilizes special functions to simplify the process.

Absolute Value Inequalities

Absolute value inequalities involve expressions within absolute value signs and often represent a range of solutions. Absolute values consider the magnitude of a number, ignoring its sign, so they essentially turn negative solutions into positive ones.

For instance, consider the inequality \(|x| < 3\). It implies that the values of \(x\) are less than 3 units away from zero, covering both the positive and negative sides. As a result, it is solved as two separate inequalities:

• \(-3 < x < 3\)

This can be input into calculators for graphing, but since absolute value functions return non-negative results, the graph will always appear above the x-axis, showcasing the viable range.

Logic Menu on TI-83/84

The "Logic Menu" on the TI-83/84 calculators is a powerful tool used to input and manipulate inequalities. To access this menu, you press the "2nd" button, followed by "TEST". It presents you with options for inequality symbols, allowing you to enter conditions directly.

Here's a simple guide to using the logic menu:

• Press "2nd" and then "TEST" to open the menu.
• Select the inequality symbol you need (e.g., \(>\), \(<\), \(\geq\), \(\leq\)).
• Enter the rest of your inequality using numbers and variables as needed.

By using the Logic Menu, students can efficiently input compound and absolute value inequalities for graphing without manual arithmetic.

Graphing Calculator Functions

Graphing calculators like the TI-83/84 come equipped with a variety of functions that make solving inequalities visually straightforward. These functions include options for entering, plotting, and analyzing graph-based solutions.

Essential graphing calculator functions involve:

• Clearing previous entries: Use this function to start fresh and avoid display clutter.
• Setting the calculator to DOT mode: This adds clarity to the display by representing the graph in dots.
• Using standard viewing windows: Ensure your x and y range forms a square grid typically of \([-10,10]\) to accommodate most graphical outputs.
• Graph button: After preparing the calculator, pressing 'GRAPH' will render your graph according to the entered inequalities.

Graphing calculators help translate algebraic solutions into visual ones, offering a dual relationship between equations and their graphical interpretations.