Problem 62 - Algebra for College Students

Question

Solve each inequality and express the solution set using interval notation. \(5(x-6)-6(x+2)<0\)

Step-by-Step Solution

Answer

The solution set is \((-42, \infty)\).

Step 1: Distribute Terms Inside Parentheses

First, distribute the 5 to both terms inside the first set of parentheses: \[ 5(x-6) = 5x - 30 \] Next, distribute the -6 to both terms inside the second set of parentheses: \[ -6(x+2) = -6x - 12 \] Now, substitute these expressions back into the inequality: \[ 5x - 30 - 6x - 12 < 0 \]

Step 2: Combine Like Terms

Combine the like terms in the inequality from Step 1: \[ 5x - 6x = -x \] \[ -30 - 12 = -42 \] This simplifies the inequality to: \[ -x - 42 < 0 \]

Step 3: Isolate the Variable

Add 42 to both sides of the inequality to isolate \(-x\): \[ -x - 42 + 42 < 0 + 42 \] This simplifies to: \[ -x < 42 \] Now, multiply both sides of the inequality by -1. Remember that multiplying by a negative number reverses the inequality sign: \[ x > -42 \]

Step 4: Express in Interval Notation

The solution \( x > -42 \) is expressed in interval notation as: \[ (-42, \infty) \]

Key Concepts

Interval Notation

Interval notation is a way to express ranges of numbers as intervals, indicating which numbers are included in a set. In the context of solving inequalities, it is essential to convert your inequality solution into interval notation for clarity. For example, when you solve for the variable and determine that your inequality solution is all numbers greater than {-42}, you express this in interval notation as (-42, \infty). This means:

• The round parenthesis "(" means that {-42} is not included in the set of solutions, indicating it is a strict inequality (greater than, but not equal to).
• The infinity symbol "\(\infty\)" signifies all numbers greater than {-42}, going infinitely in that direction.

Thus, interval notation neatly summarizes the solution set in a concise and understandable way. Keep in mind that when dealing with other inequalities that include the solution number (e.g., \(x \leq 5\)), a square bracket "[" would be used instead of a parenthesis, to show that the number itself is included in the set.

Combining Like Terms

In algebra, combining like terms is an essential skill that helps simplify expressions and equations. Like terms are terms that contain the same variable raised to the same power. The process of combining them involves summing up their coefficients and simplifying the expression.

• For instance, in the expression {5x - 30 - 6x - 12}, the like terms are {5x} and {-6x} because they both have the variable "x".
• Combining {5x} and {-6x} results in {-x} as {5 - 6 = -1}.
• Similarly, the constants {-30} and {-12} are like terms and are combined to {-42} because {-30 - 12 = -42}.

Combining like terms simplifies an equation and prepares it for easier manipulation in subsequent steps. By reducing the number of terms, you gain a clearer view of your equation, which helps with accurately solving the inequality.

Isolating the Variable

Isolating the variable is a fundamental step in solving equations and inequalities. This process involves manipulating the equation until the variable of interest is alone on one side. The goal is to make the equation as simple as possible to find the solution directly.

• In the inequality {-x - 42 < 0}, the first step to isolate "x" is to eliminate the constant {-42} by adding {42} to both sides. This results in {-x < 42}.
• The final step involves dealing with the coefficient of "x". Since {-x} implies {-1} times {x}, you need to multiply both sides by {-1} to solve for "x". Remember, reversing the inequality sign is crucial here because multiplying by a negative number changes the inequality direction. The result is {x > -42}.

Always pay close attention when multiplying or dividing by negative numbers, as it reverses the inequality sign, which is a common step in isolating the variable. Successfully isolating the variable helps in accurately determining the range of solutions.