Problem 11 - The Calculus with Analytic Geometry

Question

In Exercises 11 through 32 , find the solution set of the given inequality and illustrate the solution on the real number $$ 5 x+2>x-6 $$

Step-by-Step Solution

Answer

The solution is \( x > -2 \).

Step 1: Isolate the variable

To isolate the variable, first subtract 'x' from both sides of the inequality:\[ 5x + 2 - x > x - 6 - x \]Now simplify it:\[ 4x + 2 > -6 \]

Step 2: Solve for x

Subtract 2 from both sides to further isolate the variable:\[ 4x + 2 - 2 > -6 - 2 \]This simplifies to:\[ 4x > -8 \]Now, divide both sides by 4:\[ x > -2 \]

Step 3: Illustrate the solution on the real number line

On a number line, represent the solution set by drawing a circle at -2 and shading everything to the right of -2, indicating that all numbers greater than -2 are part of the solution set.

Key Concepts

solving inequalities

Understanding and solving inequalities is fundamental in algebra. An inequality compares two expressions, showing if one is greater, less, or equal to the other in certain conditions. The process of solving an inequality is similar to solving an equation, but with a few key differences. Let's break it down step-by-step.

First, it's important to always focus on getting the variable by itself, just like in equations. This often involves operations like addition, subtraction, multiplication, or division. However, caution is needed when multiplying or dividing both sides by a negative number, as this reverses the inequality sign.

For example, in the inequality given, \(5x + 2 > x - 6\), we perform operations to isolate 'x' on one side of the inequality. This involves subtracting 'x' from both sides and then isolating 'x', we derive that 'x' must be greater than \(-2\).

isolation of variable

Isolating the variable means getting the variable 'x' alone on one side of the inequality. This allows us to solve it directly. The main goal is to simplify the inequality step by step.

In the given inequality: \(5x + 2 > x - 6\), we first subtract 'x' from both sides. This reduces the equation to: \begin{align*} 5x + 2 - x > x - 6 - x \ 4x + 2 > -6 \ \text{Next, subtract 2 from both sides:} \ 4x + 2 - 2 > -6 - 2 \ 4x > -8 \ \text{Lastly, divide by 4:} \ x > -2 onumber \text{\text{\text{The solution here is that x must be greater than -2. Through simplifying step by step, we isolate 'x', making it the focal point of the inequality.}}} \ This process, although straightforward, requires careful and consistent step-by-step operations to ensure the correct isolation of the variable.}}}

real number line representation

Representing the solution of an inequality visually on a real number line helps understand the range of possible values for 'x'.
\br> For our example \( x > -2 \), we place a circle at \( -2 \) on the number line to indicate that \( -2 \) itself is not included (an open circle).

We then shade everything to the right of \( -2 \), indicating that any number greater than \( -2 \) is a valid solution to the inequality.

This visual representation makes it easier to grasp the concept of ranges and boundaries in inequalities. The open circle signifies that the boundary value is not part of the solution, keeping things clear and precise.