Problem 72 - College Algebra with Modeling and Visualization

Question

Solve each inequality numerically. Write the solution set in set-builder or interval notation, and approximate endpoints to the nearest tenth when appropriate. $$ 1-2 x \geq 9 $$

Step-by-Step Solution

Answer

The solution is \((-\infty, -4]\).

Step 1: Isolate the variable term

To isolate the term with the variable \( x \), we start by subtracting 1 from both sides of the inequality, giving us:\[1 - 2x - 1 \geq 9 - 1\]Simplifying both sides results in:\[-2x \geq 8\]

Step 2: Solve for the variable

Next, divide both sides of the inequality by \(-2\) in order to solve for \( x \). Remember, dividing by a negative number reverses the inequality sign:\[\frac{-2x}{-2} \leq \frac{8}{-2}\]This simplifies to:\[x \leq -4\]

Step 3: Write the solution in set notation

The solution \( x \leq -4 \) can be expressed in interval notation as:\[(-\infty, -4]\]

Step 4: Verify the solution

To ensure accuracy, substitute a value less than or equal to \(-4\) into the original inequality to check:For example, if \( x = -5 \):\[1 - 2(-5) = 1 + 10 = 11 \geq 9\]This confirms our solution is correct.

Key Concepts

Numerical Solutions

Solving inequalities numerically involves isolating the variable to find the range of numbers that satisfy the inequality. In numerical terms, you perform calculations step-by-step to simplify and solve the inequality. For example, in the inequality \(1 - 2x \geq 9\), you systematically manipulate the equation:

• First, subtract 1 from both sides, simplifying the expression to \(-2x \geq 8\).
• Next, divide both sides by \(-2\). Remember that dividing or multiplying both sides of an inequality by a negative number reverses the inequality sign, yielding \(x \leq -4\).

By performing these calculations, you numerically determine the solution's value, thus pinpointing where it fits in the number line or graph. Solving inequalities numerically requires careful arithmetic operations, ensuring not to forget how operations affect inequality signs.

Set-Builder Notation

Set-builder notation is a mathematical way of describing a set by defining properties or constraints its members must satisfy. It uses a formal language format and is a concise way to communicate the solution of an inequality. In our earlier example, if the solution is \(x \leq -4\), you can express this in set-builder notation as follows:

• \(\{ x \mid x \leq -4 \}\)

Here, the vertical bar (\(|\)) means "such that," and the expression reads as "the set of all \(x\) such that \(x\) is less than or equal to \(-4\)."
Using set-builder notation is useful when you want to express solutions quickly and clearly, especially in mathematical writing or advanced algebra topics. It precisely captures the conditions of a solution in a logical format.

Interval Notation

Interval notation provides a streamlined way to express the range of values that satisfy an inequality. It is visually direct and often preferred for its clarity in illustrating where a solution falls on the number line. Using the solution \(x \leq -4\), the representation in interval notation is:

• \((-\infty, -4]\)

This expression shows that the solution includes all numbers from negative infinity up to and including \(-4\). The parenthesis \((\) indicates that infinity is not a reachable endpoint, whereas the bracket \([\) shows that \(-4\) is included in the solution.
Interval notation is ideal for visualizing solutions in graph form, allowing you to see at a glance the part of the line that fulfills the inequality's conditions.

Algebraic Manipulation

Algebraic manipulation involves changing and rearranging equations or inequalities to isolate the variable. It's a core skill in solving inequalities, ensuring steps are applied correctly to maintain the equation's balance. Here, follow these essential steps:

• Identify terms involving the variable and the constant terms separately.
• Convert operations by adding, subtracting, multiplying, or dividing in order to isolate the variable term. Remember the key rule for inequalities: multiplying or dividing both sides by a negative number reverses the inequality sign.

In the example \(-2x \geq 8\), dividing by \(-2\) leads to \(x \leq -4\) after reversing the inequality. Analyzing manipulation steps ensures you don't accidentally change the original problem's conditions.
Algebraic manipulation builds the foundation for solving various mathematical problems, developing skill in logical thinking and problem-solving.