Question

solve the following inequality. graph the solution set.
|9 - 2x| > 11

select the correct choice and, if necessary, fill in the answer box to complete your choice.

a. the solution set in interval notation is
(type your answer in interval notation. use integers or fractions for any numbers in the expression.)

b. the solution is the empty set.

graph the solution set. choose the correct graph.

a. number line from -12 to 12 with a dot at 10 and an arrow to the right

b. partial number line

c. number line from -12 to 12 with a blue line from -2 to 10

d. partial number line

e. number line from -12 to 12 with a blue line from -12 to 12

f. partial number line

g. number line from -12 to 12 with a blue line from -12 to -2 and an arrow to the left, and a blue line from 10 to 12 with an arrow to the right

h. partial number line

Explanation:

Step1: Recall absolute value inequality rule

For \(|A| > B\) (where \(B>0\)), it is equivalent to \(A > B\) or \(A < -B\). Here, the inequality is \(|9 - 2x|>11\), and \(11>0\), so we can split it into two inequalities: \(9 - 2x>11\) or \(9 - 2x < - 11\).

Step2: Solve \(9 - 2x>11\)

Subtract 9 from both sides: \(-2x>11 - 9\), which simplifies to \(-2x>2\). Divide both sides by \(-2\) (remember to reverse the inequality sign when dividing by a negative number): \(x < - 1\).

Step3: Solve \(9 - 2x < - 11\)

Subtract 9 from both sides: \(-2x < - 11 - 9\), which simplifies to \(-2x < - 20\). Divide both sides by \(-2\) (reverse the inequality sign): \(x>10\).

Step4: Write the solution in interval notation

The solution is the union of the two intervals from the two inequalities. So the solution set in interval notation is \((-\infty, - 1)\cup(10, \infty)\).

Step5: Analyze the graph options

The graph should have an open circle at \(-1\) (since \(x < - 1\), not including \(-1\)) and an arrow to the left, and an open circle at \(10\) (since \(x>10\), not including \(10\)) and an arrow to the right. Looking at the options, the graph that matches this is the one with two open circles, one at \(-1\) (left arrow) and one at \(10\) (right arrow). From the given options (though some are partially visible, the correct graph should correspond to the interval \((-\infty, - 1)\cup(10, \infty)\)).

Answer:

A. The solution set in interval notation is \((-\infty, - 1)\cup(10, \infty)\)
(For the graph, the correct one should have open circles at -1 (arrow left) and 10 (arrow right), which among the given options (like H if we assume the last option with left arrow up to -1 and right arrow from 10) but based on the interval, the first part of the multiple - choice for the graph with the two open intervals would be the one representing \(x < - 1\) and \(x>10\))