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Chapter 1: Problem 11

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

Short Answer

Expert verified
The solution is \((-\infty, -\frac{1}{2})\).

Step by step solution

01

Isolate the variable term

Start by isolating the term containing the variable. We have the inequality \(2x + 1 < 0\). To isolate \(2x\), subtract 1 from both sides of the inequality:\[2x + 1 - 1 < 0 - 1\]which simplifies to:\[2x < -1\]
02

Solve for the variable

Next, solve for \(x\) by dividing both sides of the inequality by 2. Remember to divide the entire inequality by 2:\[\frac{2x}{2} < \frac{-1}{2}\]This simplifies to:\[x < -\frac{1}{2}\]
03

Express the solution in interval notation

The solution \(x < -\frac{1}{2}\) means that \(x\) can be any number less than \(-\frac{1}{2}\). In interval notation, this is written as:\((-\infty, -\frac{1}{2})\)
04

Graph the solution set

To graph the solution set on a number line, draw an open circle at \(-\frac{1}{2}\) to indicate that \(-\frac{1}{2}\) is not included in the solution set. Then shade the region to the left of \(-\frac{1}{2}\) to show that all numbers less than \(-\frac{1}{2}\) are included.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Interval Notation
Interval notation is a way of writing subsets of the real number line. It provides a concise way to convey the range of numbers that satisfy a particular condition.
For understanding interval notation, always remember:
  • Parentheses \( ( \, \ ) \) are used when an endpoint is not included in the set, indicating an open interval.
  • Brackets \( [ \, \ ] \) are used when an endpoint is included, indicating a closed interval.
  • Infinity \( \infty \) is always paired with a parenthesis because it is not a number that can be "reached," implying openness.
In the inequality \(x < -\frac{1}{2}\), \(-\frac{1}{2}\) is an endpoint that is not included, so we use a parenthesis. The solution extends infinitely to the left, so we denote this with \(-\infty\).
Therefore, the interval notation for the solution is \((-\infty, -\frac{1}{2})\). Trust that this simple notation will help accurately convey complex sets of numbers.
Solution Set
A solution set is a collection of all solutions of a given inequality or equation. For the inequality \(2x + 1 < 0\), we found the solution to be \(x < -\frac{1}{2}\). This simplifies to creating a set of numbers that are all less than \(-\frac{1}{2}\), and it's vital to understand that each element in this set satisfies the inequality when substituted back into the original equation.
Imagine testing number values in this set:
  • Numbers like \(-1\), \(-2\), or even \(-10\) all work because they are less than \(-\frac{1}{2}\).
  • However, numbers like \(-\frac{1}{4}\) or \(0\) do not, as they exceed \(-\frac{1}{2}\).
By applying this methodical approach, you not only establish the solution set but reinforce your grasp of how inequalities function. Remember, the solution set is essentially a narrative of all possible answers.
Number Line Graph
The number line graph is a visual representation of the solution set. It's a simple yet powerful tool to understand and express inequalities. For our inequality \(x < -\frac{1}{2}\), we need a number line to illustrate this.
Here's how you can graphically represent this interval:
  • Start by drawing a straight horizontal line 鈥� this is your number line.
  • Locate and mark the point \(-\frac{1}{2}\) on the line.
  • Since \(-\frac{1}{2}\) is not included, we use an "open circle," signaling it's not part of the solution set.
  • Shade the line to the left of this open circle to indicate that numbers less than \(-\frac{1}{2}\) are solutions.
This visual aids in quickly identifying the range of solutions an inequality offers. It solidifies the transition from algebraic to graphical interpretation, making abstract solutions tangible.

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