Problem 46 - Algebra for College Students

Question

Solve each inequality and express the solution set using interval notation. -2(3 x+2) \leq 18

Step-by-Step Solution

Answer

The solution set is \([-\frac{11}{3}, \infty)\).

Step 1: Distribute the -2

First, distribute the -2 across the terms inside the parentheses. This means you multiply -2 by both 3x and 2: \[ -2(3x + 2) = -6x - 4 \] This simplifies the inequality to: \[ -6x - 4 \leq 18 \]

Step 2: Move Constant Term to the Other Side

Add 4 to both sides of the inequality to isolate the terms with \(x\) on one side:\[ -6x - 4 + 4 \leq 18 + 4 \] Simplifying, you get: \[ -6x \leq 22 \]

Step 3: Solve for x

Divide both sides of the inequality by -6. Remember that when dividing both sides of an inequality by a negative number, the inequality sign flips:\[ x \geq \frac{22}{-6} \] Simplifying the fraction, we get:\[ x \geq -\frac{11}{3} \]

Step 4: Express the Solution in Interval Notation

The solution to the inequality \( x \geq -\frac{11}{3} \) is expressed in interval notation as:\[ \left[ -\frac{11}{3}, \infty \right) \]

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows you to simplify expressions by multiplying each term inside a parenthesis by an outside factor.
This property is often represented as:

• If you have an expression like \( a(b + c) \), the distributive property allows you to write it as \( ab + ac \).

In the context of inequalities, like in the original exercise, applying the distributive property correctly is crucial for simplifying the problem into a more manageable form.
In our example, the expression \(-2(3x + 2)\) employs the distributive property to transform into \(-6x - 4\).
This step ensures the inequality becomes easier to solve and sets the stage for the next steps in the solution.

Interval Notation

Interval notation is a mathematical way of representing a range of numbers, particularly when discussing solution sets for inequalities.
It denotes the start and end of an interval.
The intervals can either be bounded or unbounded:

• A bounded interval has both a starting and an ending point, like \([a, b]\).
• An unbounded interval extends indefinitely in one direction, such as \([-\infty, b]\) or \([a, \infty)\).

In the original exercise, the solution to the inequality \( x \geq -\frac{11}{3} \) is expressed in interval notation as \([-\frac{11}{3}, \infty)\).
Here, the square bracket \([\,]\) indicates -11/3 is included in the solution set, while the parenthesis \((\,\,)\) on infinity indicates that it is not a definite number but a concept extending indefinitely.

Solving Inequalities

Solving inequalities involves finding the values of a variable that make the inequality true.
It's similar to solving equations, but with some additional rules:

• The direction of the inequality sign matters; it determines whether the numbers are greater than, less than, or equal to a particular value.
• Importantly, when you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. This is a critical step that many students often overlook.

In our exercise, we started with the inequality \(-6x - 4 \leq 18\).
By moving terms across the inequality and applying multiplication by a negative, we transformed it into \( x \geq -\frac{11}{3} \).
Solving inequalities not only provides insights into the range of possible solutions but also enhances problem-solving skills in algebra.