Problem 37 - Algebra and Trigonometry

Question

In all exercises other than \(\varnothing\), use interval notation to express solution sets and graph each solution set on a number line. In Exercises \(27-50,\) solve each linear inequality. $$ 2 x-11<-3(x+2) $$

Step-by-Step Solution

Answer

The solution to the inequality \(2x - 11 < -3(x+2)\) is \(x < 1\). In interval notation, this is \((-∞, 1)\).

Step 1: Simplify the Inequality

Distribute the -3 on the right side of the inequality: \(2x - 11 < -3x - 6\).

Step 2: Rearrange the Equation

The goal is to get \(x\) terms on one side of the inequality and constant terms on the other. Add \(3x\) to both sides to move \(x\) to the left side, and add \(11\) to both sides to move the constant to the right side. This gives: \(5x < 5\).

Step 3: Solve for x

In this step, divide by \(5\) on both sides, which will allow to isolate \(x\): \(x < 1\).

Step 4: Write in Interval Notation and Draw on Number Line

The solution in interval notation is \((-∞, 1)\). To represent this on a number line, place an open circle at \(1\) and shade to the left to represent all numbers less than \(1\).

Key Concepts

Interval Notation

Interval notation is a way of writing subsets of the real number line using intervals. It's a shorthand for expressing the set of numbers that fall within certain bounds. For linear inequalities like the example given, where we have found the solution to be all values of x that are less than 1, we use the notation \( (-\infty, 1) \). This tells us that our solution includes all numbers up to but not including 1.
The symbols used in interval notation are important:

• \( (a, b) \) denotes all numbers between a and b, but not including a or b.
• \( [a, b] \) includes the endpoints, meaning a and b are solutions too.
• \( (-\infty, a) \) or \( (a, \infty) \) are used when the solutions extend indefinitely in the negative or positive direction, respectively. \infty symbolizes infinity and is never included in the interval (hence always denoted with a parenthesis).

These conventions are key to clearly communicating the range of solutions to inequalities.

Number Line Graphing

Graphing on a number line offers a visual representation of all possible solutions to an inequality. In the exercise provided, once the inequality is solved and we discover that x is less than 1, we illustrate this solution by drawing a number line.
Here's how:

• Draw a horizontal line with numbers marked at intervals.
• Locate the number 1 on this line.
• Place an open circle on the number 1 to show that it is not part of the solution set.
• Shade the line left of the open circle to indicate that all numbers up to, but not including, 1 are part of the solution.

The open circle is crucial because it shows that the endpoint is not included in the solution. If the endpoint were part of the solution, we'd use a closed circle instead. This visual aid complements the interval notation and provides a clear and immediate understanding of the set of solutions.

Linear Inequality Manipulation

To solve linear inequalities, we manipulate them in ways similar to how we solve equations, with some caution regarding the direction of the inequality. When we look at the initial problem, \(2x - 11 < -3(x + 2)\), our objective is to isolate x to one side. However, unlike equations, when we multiply or divide by a negative number, we must reverse the inequality sign.
A step-by-step manipulation might include:

• Distribute any terms inside parentheses.
• Get all the x variables on one side by adding or subtracting them from both sides.
• Move constant terms to the opposite side via addition or subtraction.
• If needed, divide by the coefficient of x keeping in mind the sign flipping rule.

As illustrated in the given solution steps, being methodical in the manipulation process helps to avoid errors and leads to the correct interval notation and number line graphing representation.