Question

Solve each inequality. Express your answer using set notation or interval notation. Graph the solution set. $$ |3 t-2| \leq 4 $$

Short answer

The solution set is \[ [ \frac{-2}{3}, 2 ] \]

Step-by-step solution

Understand the Absolute Value Inequality

The expression \(|3t - 2| \leq 4\) means that the distance between \(3t - 2\) and 0 must be less than or equal to 4. This can be broken into two separate inequalities.

Break into Two Inequalities

The absolute value inequality \(|3t - 2| \leq 4\) is equivalent to \[ -4 \leq 3t - 2 \leq 4 \].

Solve the Compound Inequality - Isolate the Variable

First, add 2 to all parts of the inequality: \[ -4 + 2 \leq 3t - 2 + 2 \leq 4 + 2 eq -2 \leq 3t \leq 6 . \]

Solve the Compound Inequality - Divide by 3

Next, divide all parts of the inequality by 3: \[ \frac{-2}{3} \leq t \leq 2 . \]

Express Solution Using Interval Notation

The solution to the inequality can be written in interval notation as: \[ [ \frac{-2}{3} , 2 ] \]

Graph the Solution Set

On a number line, graph the interval: start at \( \frac{-2}{3} \) and end at 2, both endpoints are included, so use solid dots.

Key concepts

absolute value inequality

Let's start by understanding what an absolute value inequality is. An absolute value inequality involves an expression within absolute value symbols, like \( |x| \), and relates it to a number using inequality signs \( <, \leq, >, \geq \). The absolute value \(|x|\) measures the distance of \(x\) from 0 on the number line.
For example, in \(|3t - 2| \leq 4\), we're looking for all \(t\) such that the expression \(3t - 2\) is at most 4 units away from 0.
This inequality can be interpreted in two ways:

• \(3t - 2\) is less than or equal to 4
• \(3t - 2\) is greater than or equal to -4

Combining these, you get a compound inequality: \( -4 \leq 3t - 2 \leq 4 \).

interval notation

Interval notation is a way of writing subsets of the real number line. It is useful for expressing the solution to inequalities. In interval notation, brackets and parentheses are used to include or exclude endpoints of the intervals.

• \([a, b]\) means that the interval includes \(a\) and \(b\)
• \((a, b)\) means that the interval does not include \(a\) and \(b\)
• \([a, b)\) or \((a, b]\) means the interval includes only one endpoint

For our inequality \(-\frac{2}{3} \leq t \leq 2\), the interval notation is \(\left[ -\frac{2}{3}, 2 \right]\). This indicates all values of \(t\) between and including \(-\frac{2}{3}\) and 2.

compound inequality

A compound inequality combines two inequalities that are joined by 'and' (\(\&\&\)) or 'or' (\(||\)). For example:

• \(-4 \leq 3t - 2 \leq 4\) is a compound inequality because it consists of two inequalities: \(3t - 2 \leq 4\) and \(-4 \leq 3t - 2\)

Solving a compound inequality involves working with both parts simultaneously.
In our exercise, we split the absolute value inequality into: \(-4 \leq 3t - 2 \leq 4\)
We solved this by isolating \(t\). Initially, adding 2 to all parts: \(-4 + 2 \leq 3t - 2 + 2 \leq 4 + 2\), simplifying to: \(-2 \leq 3t \leq 6\). Next by dividing by 3: \(\frac{-2}{3} \leq t \leq 2\), giving us the solution.

graphing inequalities

Graphing inequalities involves representing the solution set on a number line. Using solid or open dots, and shading parts of the line:

• Solid dots \((\bullet)\) represent numbers that are included in the solution (\(\leq\), \(\geq\))
• Open dots \((\circ)\) represent numbers that are not included (\(<\), \(>\))

For \(\frac{-2}{3} \leq t \leq 2\), you place solid dots on \(-\frac{2}{3}\) and 2 on the number line because they are included in the solution. Then, shade the line between them to show all values of \(t\) that satisfy the inequality.

solving linear inequalities

Linear inequalities are similar to linear equations, except they have an inequality sign. Solving them involves similar steps to solving linear equations but requires careful handling of inequality rules:

• Adding or subtracting the same number from both sides does not change the inequality direction.
• Multiplying or dividing both sides by a positive number does not change the inequality direction.
• Multiplying or dividing both sides by a negative number reverses the inequality direction.

In our example, we handled the linear inequality \(-4 \leq 3t - 2 \leq 4\) by adding and dividing appropriately:
1) Add 2 to each part, \(-4 + 2 \leq 3t \leq 4 + 2\), which simplifies to \(-2 \leq 3t \leq 6\)
2) Divide each part by 3, \(\frac{-2}{3} \leq t \leq 2\), giving us the final solution.