Problem 19 - Calculus with Concepts in Calculus

Question

Solve the inequality. $$ -6 x-2>5 $$

Step-by-Step Solution

Answer

The solution is: \(x < -\frac{7}{6}\).

Step 1: Isolate the variable term

Start by isolating the term with the variable on one side of the inequality. To do this, add 2 to both sides of the inequality:\(-6x - 2 + 2 > 5 + 2\) This simplifies to:\(-6x > 7\)

Step 2: Solve for the variable

Divide both sides of the inequality by -6. Remember, when you divide or multiply both sides of an inequality by a negative number, you must reverse the inequality sign.\(\frac{-6x}{-6} < \frac{7}{-6} \)The inequality now becomes:\(x < -\frac{7}{6}\)

Key Concepts

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. These symbols can represent numbers or operations in expressions and equations. An important aspect of algebra is solving expressions or equations to find the value of unknown variables.

When working with inequalities, algebra helps us express a range of values rather than a single solution. It uses inequality symbols like '>', '<', '≥', and '≤' to show relationships between two expressions. For example, the inequality \(-6 x - 2 > 5\) implies that there is a range of values that \(x\) can satisfy in order for the inequality to hold true.

Algebraic manipulations such as adding, subtracting, multiplying, or dividing are key operations. They help isolate variables and solve both equations and inequalities. Each step follows algebraic rules to maintain the balance and relationship expressed by the inequality.

Solving Inequalities

Solving inequalities involves finding the set of all possible values of the variable that make the inequality true. Unlike equations, where the solution is a particular value, inequalities present a range of values.

To solve an inequality such as \(-6x - 2 > 5\), we start by simplifying and isolating the variable. The steps follow logical rules similar to solving equations, with a crucial difference: whenever we multiply or divide by a negative number, we must reverse the inequality sign.

Here's a helpful list of steps when solving inequalities:

• Add or subtract quantities to isolate the term with the variable on one side.
• Multiply or divide to solve for the variable, remembering to flip the inequality sign if you multiply or divide by a negative number.
• Double-check the solution by plugging numbers from your solution set to verify they make the original inequality true.

Solving inequalities provides valuable practice as it strengthens understanding of algebraic manipulation and logical reasoning.

Variable Isolation

Variable isolation is one of the foundational techniques in algebra that aids in solving both equations and inequalities. The primary goal is to get the variable by itself on one side of the inequality or equation so that we can determine its possible values.

For instance, in the inequality \(-6x - 2 > 5\), the first step to isolate the variable \(x\) involves eliminating other numbers from its side. We start by adding 2 to both sides, resulting in \(-6x > 7\). This moves towards having \(x\) on its own.

To complete the isolation, dividing both sides of the inequality by \(-6\) is necessary. Importantly, dividing by a negative number requires flipping the inequality sign. Thus, \(x < -\frac{7}{6}\) is the solution.

Remembering this rule about reversing the inequality sign when dividing or multiplying by a negative number is pivotal in solving inequalities correctly. Proper isolation of variables thus allows for accurate and efficient problem-solving in algebra.