Question

Solve each inequality. Express your answer using set notation or interval notation. Graph the solution set. \(x+1<5\)

Short answer

x < 4. Interval notation: (-∞, 4)

Step-by-step solution

Subtract 1 from both sides

To solve the inequality, first isolate the variable by subtracting 1 from both sides of the inequality: x + 1 - 1 < 5 - 1

Simplify the inequality

Simplify both sides: x < 4

Express the solution in interval notation

The solution set for the inequality is all real numbers less than 4. In interval notation, this is written as: (- ∞, 4)

Graph the solution set

To graph this on a number line, draw an open circle at 4 and shade everything to the left of it, indicating all values less than 4 are included in the solution set.

Key concepts

Set Notation.

Set notation is a helpful way to understand the solutions of inequalities. A set is a collection of items, which in this case are numbers that satisfy the inequality.
For the inequality \(x + 1 < 5\), we solved it to find \(x < 4\).
In set notation, this is written as:
\(\{ x | x < 4 \}\).
The curly braces \(\{\}\) enclose the solution set. The vertical bar '|' means 'such that'. So, \(\{ x | x < 4 \}\) reads as 'the set of all x such that x is less than 4'.
Set notation is very powerful for depicting the grouping of numbers that fulfill the inequality.

Interval Notation.

Interval notation provides a compact way to write the continuous range of solutions for inequalities. It's very useful for describing the span of numbers between boundaries.
The solution for \(x + 1 < 5\) simplifies to \(x < 4\).
In interval notation, this is written as:
\((-\infty, 4)\).
Here's how it works:

• The parentheses \((\text{or} )\) are used to indicate that the endpoint is not included in the set (it's called an 'open interval').
• The symbol \(-\infty\) (negative infinity) means the set extends indefinitely to the left.

So, \((-\infty, 4)\) means all real numbers less than 4, from negative infinity up to, but not including, 4.
Interval notation is a very neat and concise way to represent solution sets.

Graphing Inequalities.

Graphing inequalities on a number line visually represents the solution set. This helps to intuitively understand which numbers satisfy the inequality.
To graph \(x < 4\):

• First, draw an open circle at 4.
• An open circle shows that 4 is not included in the solution set. If 4 were included (e.g., \(x \leq 4\)), we would use a closed circle.
• Next, shade the number line to the left of 4.

The shaded area signifies all numbers less than 4 are included in the solution set.
This visual aid makes it easier to see which values work for the inequality, providing a comprehensive view of the solution.
Graphing inequalities is a great way to cross-check your interval or set notation answers.