Problem 85 - Intermediate Algebra

Question

Use a graphing calculator to solve each inequality. Write the solution set using interval notation. See Using Your Calculator: Solving Linear Inequalities in One Variable. $$ 2 x+3<5 $$

Step-by-Step Solution

Answer

The solution set in interval notation is \((-\infty, 1)\).

Step 1: Rearrange the Inequality

To solve the inequality \(2x + 3 < 5\), start by isolating the variable term on one side. Subtract 3 from both sides to get \(2x < 2\).

Step 2: Solve for x

Next, divide both sides of the inequality by 2 to solve for \(x\). This gives us \(x < 1\).

Step 3: Use Interval Notation

To express the solution in interval notation, identify all values that satisfy \(x < 1\). This is expressed as \((-\infty, 1)\).

Key Concepts

Graphing Calculator

Graphing calculators are powerful tools for visualizing mathematical concepts and helping to solve inequalities. They are especially useful in solving linear inequalities by allowing you to see the solution as a graph, thus making it easier to understand the range of solutions. Many graphing calculators have a function that can directly graph inequalities. To utilize this tool:

• Enter the inequality into the graphing calculator. For instance, in our case, input "y = 2x + 3" as if it was an equation.
• Graph the line by plotting the equation and then visualize where the graph lies below or above the horizontal line "y = 5" on a 2D coordinate system.
• By evaluating where the line is strictly less than or greater than a constant, you'll see the solution set. In our example, "2x + 3 < 5" shows the region under the horizontal line at "y = 5".

The intersection or shading beneath that line reveals the valid set of solutions.

Interval Notation

Interval notation is a concise way to express solutions to inequalities. It describes the range of values that satisfy the inequality, indicating the set of all numbers between a starting point and an endpoint.In the inequality solution from our example, where we found that "x < 1", interval notation expresses this solution set efficiently:

• The notation \((-infty, 1)\) signifies that the solution includes all values less than 1 but does not include 1 itself.
• An open parenthesis "(" is used since 1 is not part of the solution set (indicating that the boundary is not included).
• \(-\infty\) always uses an open parenthesis since infinity isn’t a number you can reach.Interval notation gives a quick snapshot of the range without listing all individual numbers.

One Variable Inequalities

One variable inequalities describe a range of values that a single variable can take. They are similar to equations but instead of stating values that make two sides equal, they describe a set of possible solutions where one side is greater or less than the other. To solve such inequalities:

• Start by transforming them into a simpler form using basic arithmetic operations without changing the direction of the inequality (except when multiplying or dividing by a negative number, wherein the inequality sign is reversed).
• For the inequality "2x + 3 < 5" we first subtract 3, getting "2x < 2".
• Divide both sides by 2 to isolate the variable, resulting in "x < 1".

This process shows how the solutions can be determined through straightforward arithmetic manipulations, highlighting the range of values that fulfill the inequality condition.