Question

solving and graphing
draw a graph for each inequality.

1. $x < 5$
2. $-n > 0$
3. $5 < r$
4. $-n \geq 6$
5. $-n > 6$
6. $-4 \leq x$
7. $-3 > -m$
8. $4 < n$

Explanation:

Step1: Determine the critical point

For \(x < 5\), the critical point is \(x = 5\). Since the inequality is strict (\(<\)), we use an open - circle at \(x = 5\).

Step2: Determine the direction of the graph

Since \(x\) is less than \(5\), we draw an arrow to the left of \(5\) on the number line.

For \(-n>0\), multiply both sides by \(- 1\). When we multiply an inequality by a negative number, the direction of the inequality sign changes. So \(n < 0\).

Step1: Determine the critical point

The critical point is \(n = 0\). Since the inequality is strict (\(<\)), we use an open - circle at \(n = 0\).

Step2: Determine the direction of the graph

Since \(n\) is less than \(0\), we draw an arrow to the left of \(0\) on the number line.

For \(5 < r\) (which can be written as \(r>5\)):

Step1: Determine the critical point

The critical point is \(r = 5\). Since the inequality is strict (\(>\)), we use an open - circle at \(r = 5\).

Step2: Determine the direction of the graph

Since \(r\) is greater than \(5\), we draw an arrow to the right of \(5\) on the number line.

For \(-n\geq6\), multiply both sides by \(-1\). When we multiply an inequality by a negative number, the direction of the inequality sign changes. So \(n\leq - 6\).

Step1: Determine the critical point

The critical point is \(n=-6\). Since the inequality is non - strict (\(\leq\)), we use a closed - circle at \(n = - 6\).

Step2: Determine the direction of the graph

Since \(n\) is less than or equal to \(-6\), we draw an arrow to the left of \(-6\) on the number line.

For \(-n>6\), multiply both sides by \(-1\). When we multiply an inequality by a negative number, the direction of the inequality sign changes. So \(n < - 6\).

Step1: Determine the critical point

The critical point is \(n=-6\). Since the inequality is strict (\(<\)), we use an open - circle at \(n = - 6\).

Step2: Determine the direction of the graph

Since \(n\) is less than \(-6\), we draw an arrow to the left of \(-6\) on the number line.

For \(-4\leq x\) (which can be written as \(x\geq - 4\)):

Step1: Determine the critical point

The critical point is \(x=-4\). Since the inequality is non - strict (\(\geq\)), we use a closed - circle at \(x = - 4\).

Step2: Determine the direction of the graph

Since \(x\) is greater than or equal to \(-4\), we draw an arrow to the right of \(-4\) on the number line.

For \(-3 > - m\), multiply both sides by \(-1\). When we multiply an inequality by a negative number, the direction of the inequality sign changes. So \(3 < m\) (or \(m>3\)).

Step1: Determine the critical point

The critical point is \(m = 3\). Since the inequality is strict (\(>\)), we use an open - circle at \(m = 3\).

Step2: Determine the direction of the graph

Since \(m\) is greater than \(3\), we draw an arrow to the right of \(3\) on the number line.

For \(4 < n\) (which can be written as \(n>4\)):

Step1: Determine the critical point

The critical point is \(n = 4\). Since the inequality is strict (\(>\)), we use an open - circle at \(n = 4\).

Step2: Determine the direction of the graph

Since \(n\) is greater than \(4\), we draw an arrow to the right of \(4\) on the number line.

Answer:

1. Open - circle at \(x = 5\), arrow to the left.
2. Open - circle at \(n = 0\), arrow to the left.
3. Open - circle at \(r = 5\), arrow to the right.
4. Closed - circle at \(n=-6\), arrow to the left.
5. Open - circle at \(n=-6\), arrow to the left.
6. Closed - circle at \(x=-4\), arrow to the right.
7. Open - circle at \(m = 3\), arrow to the right.
8. Open - circle at \(n = 4\), arrow to the right.