Solve the inequality both algebraically and graphically. Draw a number line graph of the solution and give interval notation.
StartFraction x minus 5 Over 3 EndFraction
less thanStartFraction 22 Over 5 EndFraction
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Part 1
The solution as an inequality is

We are tasked with solving the inequality:

\[
\frac{x - 5}{3} < \frac{22}{5}
\]

### Algebraic Solution

Step 1: Eliminate the fractions by multiplying both sides of the inequality by the least common denominator (LCD) of 3 and 5, which is 15.

\[
15 \cdot \frac{x - 5}{3} < 15 \cdot \frac{22}{5}
\]

Step 2: Simplify both sides:

\[
5(x - 5) < 3 \cdot 22
\]

This simplifies to:

\[
5(x - 5) < 66
\]

Step 3: Expand the left-hand side:

\[
5x - 25 < 66
\]

Step 4: Add 25 to both sides to isolate the term with \(x\):

\[
5x < 91
\]

Step 5: Divide both sides by 5:

\[
x < \frac{91}{5}
\]

Thus, the solution as an inequality is:

\[
x < \frac{91}{5}
\]

### Interval Notation

The solution in interval notation is:

\[
(-\infty, \frac{91}{5})
\]

### Graphical Representation

We can represent the solution on a number line. The number \( \frac{91}{5} = 18.2 \), so the solution represents all values of \(x\) less than 18.2. The point at \(x = 18.2\) will be an open circle to indicate that the value 18.2 is **not** included in the solution.

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Would you like a number line graph representation for this inequality or further explanation?

### 5 Related Questions:
1. How do you solve inequalities involving absolute values?
2. What changes when multiplying or dividing an inequality by a negative number?
3. How would you graph compound inequalities on a number line?
4. How is the graphical solution of a quadratic inequality different from a linear one?
5. Can inequalities have no solutions? If so, how?

### Tip:
When solving inequalities, always be cautious when multiplying or dividing by negative numbers—remember to flip the inequality sign!

Solve the inequality both algebraically and graphically. Draw a number line graph of the solution and give interval notation. StartFraction x minus 5 Over 3 EndFraction less thanStartFraction 22 Over 5 EndFraction.

Learn how to solve the inequality (x - 5)/3 < 22/5 both algebraically and graphically with step-by-step instructions. This solution covers multiplying to eliminate fractions, simplifying expressions, and representing the solution on a number line. Suitable for students in Grades 9-11.

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