Graph -n< 2 or 2n-3> 5
The presented problem involves inequalities with a variable n and asks for a graphical representation of the solutions. There are two separate inequalities to consider:
-n < 2 states that negative n is less than 2, which you would interpret and solve for n.
2n - 3 > 5 represents a condition where two times n minus three is greater than five, which again you would solve for n.
The question involves creating a single graph that visually represents all values of n that satisfy either of the two inequalities. Because the problem uses the word "or," it indicates that any value that satisfies at least one of the inequalities should be included in the solution set on the graph.
$- n < 2$or$2 n - 3 > 5$
Answer
Solution:
Step 1:
Plot the inequalities on a number line. $-n < 2$ or $2n - 3 > 5$
Step 2:
First, solve the inequality $-n < 2$.
Multiply both sides by $-1$ and reverse the inequality sign. $n > -2$
Step 3:
Next, solve the inequality $2n - 3 > 5$.
Add $3$ to both sides to isolate the term with $n$. $2n > 8$ Divide both sides by $2$ to solve for $n$. $n > 4$
Step 4:
Graph the solutions on a number line. For $n > -2$, shade the number line to the right of $-2$. For $n > 4$, shade the number line to the right of $4$.
Step 5:
Since the original statement is an "or" statement, combine the shaded regions on the number line. The solution set includes all numbers greater than $-2$ and all numbers greater than $4$, which is effectively all numbers greater than $-2$.
Knowledge Notes:
To solve and graph inequalities, one must understand the following key concepts:
Inequality Symbols: The symbols $<$, $>$, $\leq$, and $\geq$ are used to denote less than, greater than, less than or equal to, and greater than or equal to, respectively.
Solving Inequalities: Similar to solving equations, but with an important difference: multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign.
Graphing Inequalities: This involves representing the solution set on a number line. For strict inequalities ($<$ or $>$), an open circle is used to indicate that the endpoint is not included. For inclusive inequalities ($\leq$ or $\geq$), a closed circle is used to indicate that the endpoint is included.
Compound Inequalities: These are two or more inequalities joined by "and" or "or". An "and" compound inequality requires that both inequalities be true at the same time, resulting in the intersection of the solution sets. An "or" compound inequality requires that at least one of the inequalities be true, resulting in the union of the solution sets.
Number Line: A visual representation of numbers in order, typically used to display the solution set of an inequality. Positive numbers are to the right of zero, and negative numbers are to the left.
In the given problem, we have a compound inequality with an "or" statement, which means we look for the union of the solution sets of the individual inequalities.
