Question

Solve each inequality. Express your answer using set notation or interval notation. Graph the solution set. \(0<1-\frac{1}{3} x<1\)

Short answer

The solution is \((0, 3)\) and can be graphed as an open interval between 0 and 3 on a number line.

Step-by-step solution

Understand the Inequality

The inequality to solve is given as a compound inequality: \[0 < 1 - \frac{1}{3}x < 1\]

Break Down the Compound Inequality

First, split the compound inequality into two separate inequalities:\[0 < 1 - \frac{1}{3}x\]and\[1 - \frac{1}{3}x < 1\]

Solve the First Inequality

Solve the first part of the inequality:\[0 < 1 - \frac{1}{3}x\]Subtract 1 from both sides:\[-1 < -\frac{1}{3}x\]Multiply both sides by -3 (and reverse the inequality sign):\[3 > x\] which simplifies to \[x < 3\]

Solve the Second Inequality

Solve the second part of the inequality:\[1 - \frac{1}{3}x < 1\]Subtract 1 from both sides:\[ - \frac{1}{3}x < 0\]Multiply both sides by -3 (and reverse the inequality sign):\[ x > 0\]

Combine the Solutions

Combine the solutions from the two inequalities:\[0 < x < 3\]

Express the Solution in Interval Notation

The solution in interval notation is:\((0, 3)\)

Graph the Solution

Graph the interval \((0, 3)\) on a number line by drawing an open circle at 0 and 3, and shading the region between them.

Key concepts

Understanding Compound Inequalities

When dealing with compound inequalities, you are working with two separate inequalities at the same time. These inequalities are typically connected by the word 'and' or 'or.' In our exercise, the compound inequality is written as: \[0 < 1 - \frac{1}{3}x < 1\] This 'and' compound inequality suggests that both conditions must be met simultaneously.
Compound inequalities with 'and' require finding values that satisfy both inequalities at the same time. Breaking down the compound inequality into two standalone parts helps to solve it step by step.
First, split the inequality into two separate ones: \[0 < 1 - \frac{1}{3}x\] and \[1 - \frac{1}{3}x < 1\] By solving each part independently, you find the common values that satisfy both conditions.

Inequality Notation Explained

Inequality notation helps to express the range of values that a variable can take. In our example, we encounter inequalities like \[0 < 1 - \frac{1}{3}x < 1\] which means that the expression \[1 - \frac{1}{3}x\] has to be greater than 0 but less than 1.
To solve it, you perform operations to isolate the variable while respecting the direction of the inequality. For example:

• Subtracting numbers from both sides.
• Multiplying or dividing by negative numbers, which reverses the inequality sign.

Let's take \[0 < 1 - \frac{1}{3}x\]and solve it step-by-step:

• Subtract 1 from both sides: \[-1 < - \frac{1}{3}x\]
• Multiply both sides by -3 and reverse the inequality: \[3 > x\]

Thus, \[x < 3\].
Similarly, solve the second part \[1 - \frac{1}{3}x < 1\]:

• Subtract 1: \[- \frac{1}{3}x < 0\]
• Multiply by -3 and reverse inequality: \[x > 0\]

Interval Notation Basics

Interval notation provides a compact way to represent ranges of values. It's useful for defining solution sets of inequalities.
Interval notation uses parentheses \(( )\) and brackets \([ ]\) to show whether endpoints are included or excluded.
Parentheses mean the endpoint is not included (open), while brackets mean it is included (closed).
For example, the interval \((0, 3)\) means all values between 0 and 3, but not including 0 and 3 themselves.
To convert our solution \[0 < x < 3\] into interval notation:

• Write the lower bound: \(0\) (not included, so use a parenthesis).
• Write the upper bound: \(3\) (not included, so use a parenthesis).

Combining these, we get \((0, 3)\).

Solution Sets and Graphing

A solution set includes all possible values that satisfy an inequality. Graphing a solution set offers a visual representation, which helps to understand it better.
For our exercise, the solution set is given by \[0 < x < 3\].
Expressing this in interval notation, we have \((0, 3)\).
Graphing this on a number line:

• Draw a number line.
• Place open circles at the endpoints 0 and 3 (since they are not included in the solution set).
• Shade the region between 0 and 3 to indicate all the values x can take.

This visual helps in quickly understanding the range of potential solutions. Always check the inequality signs to know if the endpoints are included or excluded.