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Chapter 5: Problem 16

Graph the solution of each inequality on a number line. $$-5 \leq x+1 < 5$$

Short Answer

Expert verified
The solution to the compound inequality \(-5 \leq x+1 < 5\) is \(x \geq -6\) and \(x < 4\), represented on the number line from -6 to 4, including -6 but excluding 4.

Step by step solution

01

Understand Compound Inequality

First, understand that \(-5 \leq x+1 < 5\) is the same as \(-5 \leq x+1\) AND \(x+1 < 5\). This means find an x that makes both parts true.
02

Solve Each Inequality

For \(-5 \leq x+1\), subtract 1 to isolate x giving \(x \geq -6\) (flipped the inequality because it was a 'greater than' or 'equal to' form). For \(x+1 < 5\), again subtract 1 to isolate x giving \(x < 4\). So the solution to the compound inequality is \(-6 \leq x < 4\).
03

Graph the Solution

On the number line, the graph starts at -6 and goes up to but does not include 4. At -6, place a solid dot or bracket ( [ ] symbol) to indicate that -6 is part of the solution. At 4, place an open dot or parenthesis ( ( ) symbol) to indicate that 4 is not part of the solution. Fill in the space between -6 and 4 to show all the numbers that are part of the solution.

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Key Concepts

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

Number Line
The number line is a visual representation of numbers laid out in a straight line. It covers all numbers鈥攑ositive and negative, whole numbers, fractions, and decimals鈥攑ositioned according to their value. Each number on the number line corresponds to a point.

When working with inequalities, the number line becomes a handy tool to display the range of solutions visually. For example, if we want to illustrate the solution for \(-6 \leq x < 4\), we would represent it on this line.
  • The range of numbers is shown by marking specific points accompanied by solid or open dots.
  • Solid dots or brackets indicate that the number is inclusive鈥攖hat is, part of the solution.
  • Open dots or parentheses are used to show that a specific number itself is not part of the solution.
This visual tool makes it easier to discern which values satisfy the inequality.
Compound Inequalities
Compound inequalities consist of two inequalities joined by "and" or "or." They specify a range of possible solutions by combining multiple conditions. The given exercise, \(-5 \leq x+1 < 5\), illustrates a compound inequality connected by "and." Here are some key things to remember:

  • With the "and" compound inequality, both conditions must be true simultaneously.
  • To solve, split the inequality into two parts: \(-5 \leq x+1\) and \(x+1 < 5\). Simplify each to isolate \(x\).
  • Combine the simplified parts to obtain a solution set that satisfies both conditions, such as \(-6 \leq x < 4\).
The final solution set denotes the segment of values that all meet the initial inequality terms. Compound inequalities allow us to express a range of numbers that fit those specified criteria.
Graphing Inequalities
Graphing inequalities involves shading a region on a number line to show all possible solutions for an inequality. For the compound inequality \(-6 \leq x < 4\), follow these steps to graph it accurately:

  • Identify the boundary points, which are -6 and 4 in this case.
  • Place a solid dot or bracket at -6 to signal that -6 is part of the solution set due to the "less than or equal to" condition.
  • Place an open dot or parenthesis at 4 to indicate that it is not part of the solution, aligning with the "less than" condition.
  • Shade or draw a line between these points to include all numbers between -6 and 4. This shading visually represents the solution set of the inequality on the number line.
Graphing provides a clear, direct way of interpreting which values satisfy the inequality, making it simpler to understand the relationship between those numbers.

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Most popular questions from this chapter

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