Problem 17 - Algebra and Trigonometry

Question

Solve the linear inequality. Express the solution using interval notation and graph the solution set. $$ 2 x+1<0 $$

Step-by-Step Solution

Answer

x < -0.5 or \((-\infty, -0.5)\)

Step 1: Understand the Inequality

The inequality given is \(2x + 1 < 0\). The goal is to solve for \(x\) such that the inequality holds true.

Step 2: Isolate the Linear Term

Subtract 1 from both sides of the inequality to move the constant to the right side: \(2x + 1 - 1 < 0 - 1\), simplifying to \(2x < -1\).

Step 3: Solve for x

Divide both sides by 2 to solve for \(x\): \(2x/2 < -1/2\) which simplifies to \(x < -0.5\).

Step 4: Express in Interval Notation

The inequality \(x < -0.5\) in interval notation is \((-\infty, -0.5)\).

Step 5: Graph the Solution Set

On a number line, open circle at \(-0.5\) and shade the line to the left, indicating that all numbers less than \(-0.5\) are solutions.

Key Concepts

Solving Linear Inequalities

Linear inequalities are similar to linear equations, but instead of an equals sign, they have inequality signs like <, >, ≤ or ≥. Solving a linear inequality means finding all possible values of the variable that make the inequality true.To solve the inequality \(2x + 1 < 0\), you follow similar steps as solving a regular equation:

• Isolate the variable term on one side of the inequality.
• Perform operations like addition, subtraction, multiplication, or division, being careful to reverse the inequality sign when multiplying or dividing by a negative number.

For this inequality, we first subtract 1 from both sides, getting \(2x < -1\), and then divide by 2 to find \(x < -0.5\). This results in all values of \(x\) that are less than -0.5 satisfying the inequality.

Interval Notation

Interval notation provides a concise way to describe a range of values that satisfy an inequality. It's especially useful for expressing solutions of inequalities in a way that's easy to interpret.For the inequality \(x < -0.5\), we use interval notation to express the solution set. In interval notation, this set is written as \((-\infty, -0.5)\). Here's what this means:

• The parentheses \((\, \)) indicate that -0.5 is not included in the solution set, as the inequality is strict (\< rather than \≤).
• The symbol \(-\infty\) is used to show that the solution goes indefinitely in the negative direction.

This notation succinctly captures all possible solutions for the inequality.

Graphing Inequalities

Graphing inequalities on a number line offers a visual representation of the solution set, which can be very helpful in understanding the concept.To graph the inequality \(x < -0.5\), you follow these steps:

• First, draw a number line and locate the point -0.5.
• Place an open circle at -0.5 to indicate that -0.5 is not part of the solution (since the inequality is \< rather than \≤).
• Shade the region to the left of -0.5 to show that all values less than -0.5 are solutions to the inequality.

This graph visually represents the fact that any value of \(x\) that lies to the left of -0.5 on the number line is a valid solution.