Problem 6 - Thomas Calculus

Question

In Exercises \(5-12,\) solve the inequalities and show the solution sets on the real line. $$ 8-3 x \geq 5 $$

Step-by-Step Solution

Answer

The solution is \( x \leq 1 \), represented on the real line as a shaded line extending left from 1 with a closed circle at 1.

Step 1: Isolate the Variable

First, we need to isolate the variable on one side of the inequality to find out what values it can take. Start by subtracting 8 from both sides:\[ 8 - 3x - 8 \geq 5 - 8 \]This simplifies to:\[ -3x \geq -3 \]

Step 2: Solve for the Variable

Next, divide both sides of the inequality by -3. Remember, when dividing or multiplying an inequality by a negative number, you must flip the inequality sign:\[ \frac{-3x}{-3} \leq \frac{-3}{-3} \]This simplifies to:\[ x \leq 1 \]

Step 3: Display the Solution Set

The solution set for the inequality \( x \leq 1 \) is all real numbers less than or equal to 1. On a real number line, this is represented by a closed circle at 1 and a shaded line extending to the left, indicating all numbers less than or equal to 1.

Key Concepts

Real Number Line

The real number line is an essential tool in mathematics to visualize and understand inequalities and solutions. Think of it as an infinite straight line of numbers extending in both directions. Each point on the line corresponds to a real number.
Imagine the line as a ruler, where zero is the center point, positive numbers stretch out to the right, and negative numbers extend to the left.

• Closed Circle: A closed circle on the real number line indicates that a number is included in the solution set. For example, in our inequality, a closed circle at 1 means 1 is part of the solution.
• Open Circle: If an open circle is used, it signifies that the number is not included in the solution set.
• Shading: The shaded part of the line shows all numbers included in the inequality solution. For solutions with all numbers less than or equal to a value, like in this case, shading extends leftwards from the closed circle.

When representing solutions on this line, it's like painting a picture of all potential values that the variable can be.

Inequality Solutions

Solving inequalities involves finding all the possible values of a variable that make the inequality true. This is similar to solving equations, but with inequalities, we need to consider the range of solutions.
Inequalities differ from equations because they have more than one solution, often represented as a range or set on the real number line.
There are different types of inequalities based on their symbols:

• Greater than (">") or Less than ("<"): These inequalities suggest that the variable should be larger or smaller than a particular value.
• Greater than or equal to ("") or Less than or equal to (""): These include the specified number as part of the solution set.

The solution for inequalities doesn't stop at a single value but involves a line of connected numbers. It's crucial to display all solutions accurately on the real number line, helping to visualize their extensive range.

Algebraic Manipulation

Algebraic manipulation is the art of rearranging and simplifying equations or inequalities to find solutions effectively. It involves standard operations like addition, subtraction, multiplication, and division.
When solving inequalities, two rules are critical:

• When adding or subtracting a number, maintain the inequality direction.
• When multiplying or dividing by a negative number, always flip the inequality sign. This is because multiplying or dividing by a negative number reverses the order of values.

Remember:

• To isolate the variable, perform inverse operations. In our example, we subtracted from both sides first.
• Check your manipulation steps to avoid mistakes, especially when handling negative numbers, to ensure you don’t incorrectly flip the inequality sign.

Proper algebraic manipulation helps unlock the solution to inequalities, ensuring you accurately determine the range of possible values for the variable in question.