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Chapter 3: Problem 25

Answer the given questions. Where are all points for which \(x<-1 ?\)

Short Answer

Expert verified
The points are on the number line to the left of \(-1\), in the interval \((-\infty, -1)\).

Step by step solution

01

Understand the Inequality

The given inequality is \(x < -1\). It asks for locations of points on the number line (or in a coordinate plane) where the x-value is less than -1.
02

Visualize the Number Line or Coordinate Plane

Imagine or draw a number line extending in both directions. The position corresponding to \(x = -1\) is a reference point. In a coordinate plane, this means considering the vertical line \(x = -1\).
03

Identify the Region Satisfying the Inequality

On a number line, any point to the left of \(-1\) satisfies \(x < -1\). Similarly, in a coordinate plane, the region to the left of the line \(x = -1\) meets this condition.
04

Express the Solution Set

The solution set includes all x-values less than \(-1\). In interval notation, it is expressed as \((-\infty, -1)\). This set includes all numbers to the left of \(-1\), but does not include \(-1\) itself.

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Key Concepts

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

Number Line
A number line is a straight, horizontal line that represents all real numbers. Imagine it extending infinitely in both directions. It has a central point, usually zero, from which numbers increment positively to the right and negatively to the left. This simple visual tool is incredibly helpful for understanding inequalities like \(x < -1\). On the number line, you mark the point \(-1\) and then look to the left of it.
  • Any point to the left of \(-1\) satisfies the condition \(x < -1\).
  • Use an open circle to represent \(-1\) to show that this point is not included in the solution.
When working with number lines, remember that they help visualize differences and relationships between numbers by giving a clear picture of where values exist relative to each other. This clarity can help greatly in understanding solutions to inequalities.
Coordinate Plane
The coordinate plane is a bit like a graph with two axes: the horizontal one (x-axis) and the vertical one (y-axis). Each point on this plane is identified by an ordered pair \((x, y)\). For the inequality \(x < -1\), we focus on the x-coordinate.
  • Imagine a vertical line that intersects the x-axis at \(-1\).
  • All points on the plane to the left of this line satisfy the inequality.
Remember, the plane allows you to visualize inequalities involving two variables, though this example focuses merely on the x-variable. The coordinate plane is instrumental for graphically solving and understanding algebraic equations and inequalities.
Interval Notation
Interval notation is a way of describing a set of numbers along a number line. It's particularly useful in expressing solutions to inequalities. When writing in interval notation, we use brackets to describe the set:
  • Parentheses, \(( )\), indicate that an endpoint is not included.
  • Brackets, \([ ]\), indicate that an endpoint is included.
For the inequality \(x < -1\), the interval notation is \((-cfty, -1)\). This notation shows that the values include everything less than \(-1\), stretching infinitely but not including the value \(-1\) itself. Interval notation makes it easy to quickly see the range of values and understand the core solution to inequality problems.

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