Problem 12 - Thomas Calculus

Question

In Exercises \(5-12,\) solve the inequalities and show the solution sets on the real line. $$ -\frac{x+5}{2} \leq \frac{12+3 x}{4} $$

Step-by-Step Solution

Answer

The solution is \(x \geq -\frac{22}{5}\).

Step 1: Clear the Fractions

To clear the fractions from the inequality, we need to find the least common multiple of the denominators, which are 2 and 4. The LCM is 4. Multiply every term of the inequality by 4 to eliminate the denominators. This gives us:\[-2(x+5) \leq 12 + 3x\]

Step 2: Distribute and Simplify

Distribute the -2 on the left side of the inequality:\[-2x - 10 \leq 12 + 3x\]Now, it is in a simpler form we can solve.

Step 3: Move Variables to One Side

Add \(2x\) to both sides to get all the x-terms on one side of the inequality:\[-10 \leq 12 + 5x\]

Step 4: Isolate the Variable

Subtract 12 from both sides to get the constant terms on one side:\[-22 \leq 5x\]Now divide all terms by 5 to solve for \(x\):\[x \geq -\frac{22}{5}\]

Step 5: Represent the Solution on the Real Line

The solution \(x \geq -\frac{22}{5}\) can be represented on the real line by drawing a number line and shading the region to the right of and including \(-\frac{22}{5}\). To indicate that \(x\) can equal \(-\frac{22}{5}\), use a solid dot at \(-\frac{22}{5}\).

Key Concepts

Real Numbers

Real numbers form a continuous group that comprises all the numbers on the number line. This includes all the rational numbers (like integers and fractions) and the irrational numbers (like square roots of non-perfect squares). They are denoted as \( \mathbb{R} \). Understanding real numbers is essential in solving inequalities, as solutions can take any point along the continuum of the number line.

• **Rational Numbers:** These are numbers that can be expressed as a fraction of two integers, where the denominator is not zero (for example, \( \frac{1}{2} \), \( -6 \)).
• **Irrational Numbers:** These cannot be expressed as a simple fraction, such as \( \sqrt{2} \) and \( \pi \).

When solving inequalities, like the one we have in this exercise, the solution will be a range on the real number line rather than just a single point, due to the nature of inequalities.

Inequality Representation

Inequalities are mathematical statements that define a range of values that variables can take. They are represented using symbols:

• **\( > \) and \( < \):** Greater than and less than.
• **\( \geq \) and \( \leq \):** Greater than or equal to, and less than or equal to, respectively.

In this exercise, the inequality \(-\frac{x+5}{2} \leq \frac{12+3x}{4}\) means that the expression on the left side must be less than or equal to the expression on the right side. To solve the inequality, we aim to find the set of all \( x \) values that satisfy this relationship.
For representation purposes, solutions to inequalities can be depicted on a number line. When the solution includes the boundary value, as it does for this exercise with \( x \geq -\frac{22}{5} \), we use a solid dot. If the boundary were not included, an open circle would be used.

Step-by-Step Algebra

Solving inequalities involves a series of algebraic steps, similar to solving equations but with extra attention to the inequality sign. Each step must simplify the inequality, maintaining balance much like a balance scale.

Clearing Fractions

• The first step is to eliminate fractions by finding a common denominator. Here, this is achieved by multiplying the entire inequality by 4, simplifying the calculation process.

Distribute and Simplify

• Expand expressions as needed. For example, distribute the \(-2\) in \(-2(x+5)\) to simplify the inequality to \(-2x - 10\).

Isolating Variables

• Combine like terms and move variables to one side. This involves adding or subtracting terms to keep variables and constants organized.
• The end goal is to isolate \( x \) or the variable of interest. This results in a simplified inequality such as \(-22 \leq 5x\).

Final Division

• Divide each term by the coefficient of the variable to solve for the variable itself, ensuring the inequality maintains its direction.

By following these systematic steps, you can confidently and accurately solve inequalities.