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Chapter 2: Problem 18

Solve each of the inequalities and graph the solution set on a number line. \(x+2<1\)

Short Answer

Expert verified
The solution is \(x < -1\); graph with an open circle at \(-1\) and shade left.

Step by step solution

01

Identify the Inequality

The inequality we need to solve is \(x + 2 < 1\). Our aim is to isolate \(x\) on one side of the inequality.
02

Subtract 2 from Both Sides

To isolate \(x\), subtract 2 from both sides of the inequality. This gives \(x + 2 - 2 < 1 - 2\). Simplifying, we get \(x < -1\).
03

Graph the Solution on a Number Line

Draw a number line and mark \(-1\) on it. Since \(x\) is less than \(-1\), use an open circle on \(-1\) and shade all the numbers to the left of \(-1\) to represent \(x < -1\).

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Key Concepts

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

Graphing Inequalities
Graphing inequalities is an important part of solving and visualizing problems in algebra. When you solve an inequality, you find a range of values that satisfy it. However, visualizing these values helps to understand the solution better. To graph an inequality like \(x < -1\):
  • Start by drawing a horizontal line, which will be your number line.
  • Identify the critical number (in this case, \(-1\)) and mark it on the line.
  • Since \(x < -1\), use an open circle on \(-1\). An open circle indicates that the value \(-1\) is not included in the solution set.
  • Shade the region to the left of \(-1\) to show all values less than \(-1\). This shading indicates all potential solutions to the inequality.
Graphing in this manner helps you visually communicate the set of all solutions, making it easier to understand the scope of the inequality solution.
Number Line Representation
Number lines are a visual tool that help in representing the range of solutions for inequalities. When you use a number line:
  • You get a clear visual of the position of numbers and their relationship to each other.
  • It is easier to see how the solution extends to infinity in a particular direction.
For example, in the inequality \(x < -1\):
  • The open circle at \(-1\) clearly shows that \(-1\) is not included in the solution set.
  • The shading to the left signifies that all values less than \(-1\) are valid solutions.
This representation quickly conveys essential information about the values \(x\) can take, and helps one comprehend how solutions are distributed along the number line.
Algebraic Manipulation
Algebraic manipulation is a key step in solving inequalities, as it helps to isolate the variable. This involves using basic algebraic operations while keeping the inequality sign in mind. For the inequality \(x + 2 < 1\):
  • Start by identifying which operations will help isolate the variable \(x\). Here, subtract 2 from both sides.
  • When subtracting 2, maintain balance by performing the same operation on both sides: \(x + 2 - 2 < 1 - 2\).
  • Simplify the result to reach \(x < -1\).
  • Always remember that multiplying or dividing by a negative number flips the inequality sign. Since no such operation occurred here, the direction of the inequality remains unchanged.
By understanding these steps, you develop the skills needed to thoughtfully manipulate equations and inequalities, ensuring you derive accurate solutions that can then be effectively represented on a number line.

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Most popular questions from this chapter

For Problems \(31-54\), solve each inequality. $$ |x-3|>9 $$

For Problems \(19-34\), graph the solution set for each compound inequality, and express the solution sets in interval notation. $$ x>1 \text { and } x<4 $$

Solve each equation. $$ |x+1|=|x-1| $$

Use an algebraic approach to solve each problem. If a head of lettuce costs a retailer 0.68 dollar, at what price should it be sold to yield a profit of \(60 \%\) on the selling price?

Solve each equation. \(0.07 y=0.42\)
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