Problem 17 - Precalculus Student Solutions Manual 5th

Question

Solve each inequality. Write each solution set in interval notation. $$3(x+5)+1 \geq 5+3 x$$

Step-by-Step Solution

Answer

The solution set is (-∞, ∞).

Step 1: Distribute the 3 on the left side

First, distribute the 3 across the terms inside the parentheses on the left side of the inequality: \[ 3(x + 5) + 1 \ \rightarrow 3x + 15 + 1 \ \rightarrow 3x + 16 \ \text{Now the inequality is:} \ 3x + 16 \geq 5 + 3x \]

Step 2: Subtract 3x from both sides

Next, subtract 3x from both sides to isolate the constant term: \[ 3x + 16 - 3x \geq 5 + 3x - 3x \rightarrow 16 \geq 5 \]

Step 3: Simplify and interpret the result

Simplify the expression: \[ 16 \geq 5 \]Since this inequality is always true, it means that every real number is a solution.

Step 4: Write the solution set in interval notation

Since every real number is a solution, the solution set in interval notation is \[ (-∞, ∞) \]

Key Concepts

Inequalities

Inequalities are expressions that show the relationship between two values where one is larger or smaller than the other. They use symbols like <, >, ≤, and ≥. For example, 3x + 16 ≥ 5 + 3x means that 3x + 16 is greater than or equal to 5 + 3x. Solving inequalities often involves similar steps to solving equations, such as simplifying expressions and isolating variables. However, there are unique rules, especially when multiplying or dividing by negative numbers, as the inequality sign must be reversed in those cases.

Interval Notation

Interval notation is a way of expressing the set of solutions to an inequality. It uses brackets and parentheses to show the range of numbers included in the solution. For instance:

• Square brackets [ ] indicate that the end value is included (closed interval).
• Parentheses ( ) indicate that the end value is not included (open interval).

For the solution to the exercise where every real number satisfies the inequality, the interval notation is \((-∞, ∞)\). This means that all real numbers from negative infinity to positive infinity are included.

Distributive Property

The distributive property is a basic algebraic principle used to simplify expressions. It states that a(b + c) = ab + ac. This means you distribute the multiplier to each term inside the parentheses. In the given exercise, we apply the distributive property as follows:

• 3(x + 5) becomes 3x + 15.

This step helps break down the expression so it can be simplified further to solve the inequality.

Real Numbers

Real numbers include all rational and irrational numbers. They encompass every number you can think of along the number line, from negative infinity to positive infinity. In inequalities, when we determine the solution set involves all real numbers, it implies that any number you choose from this set will satisfy the inequality. In our exercise, since 16 ≥ 5 is always true, every real number is a solution, written in interval notation as \((-∞, ∞)\).