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Chapter 2: Problem 106

Graph the solution set, and write it using interval notation \(-3 \leq 3 x<12\)

Short Answer

Expert verified
The solution in interval notation is [-1,4) Graph it by representing the interval from -1 to 4 (excluding 4) on a number line.

Step by step solution

01

Understand the Inequality

The problem provides a compound inequality o o The compound inequality o o
02

Isolate the Variable x

To isolate x, first divide all parts of the inequality by 3.o oThis results in the simplified inequality o6
03

Write the Solution in Interval Notation

Now that the inequality is simplified, convert it to interval notation. For the interval the inner bounds are -and 6. Including -1 but excluding
04

Graph the Solution

To graph the solution on a number line:o Draw an unshadowed circle at -1 to represent x,de closed.in

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Key Concepts

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

Compound Inequalities
Compound inequalities are statements that involve two separate inequalities joined by the words 'and' or 'or'. In this problem, the given compound inequality is \(-3 \leq 3x < 12\). This combines two inequalities: \(-3 \leq 3x\) and \(3x < 12\).

When you see the word 'and', it means that both conditions must be true simultaneously. Here, x must satisfy both \(-3 \leq 3x\) and \(3x < 12\).

To solve this, we treat each part of the compound inequality separately before combining the solutions. Let's isolate x by dividing all parts of the given inequality by 3. This results in \(-1 \leq x < 4\). This tells us that x must be greater than or equal to -1 and less than 4.
Interval Notation
Once the inequality \(-1 \leq x < 4\) is found, we can express the solution in a more condensed form known as interval notation.

Interval notation uses brackets to describe the range of possible values. For instance:
  • Square brackets [ ] indicate the number is included (closed interval).
  • Parentheses ( ) indicate the number is not included (open interval).

For \(-1 \leq x < 4\), we include -1, but not 4. Thus, we write the solution in interval notation as [ -1, 4 ). Here:
  • \([ -1\) indicates -1 is included.
  • \(4 )\) indicates 4 is not included.
Graphing Inequalities
Graphing inequalities on a number line visually represents the solution range. For the inequality \(-1 \leq x < 4\), we perform the following steps:
  • Draw a number line.
  • Place a solid dot (closed circle) at -1 to show -1 is included in the solution.
  • Place an open dot (open circle) at 4 to show 4 is not included.
  • Shade the region between -1 and 4. The shading indicates all the values x can take within this range.
This graphical representation helps in understanding the range of solutions for the inequality.

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