To find which number line shows the solution set to the inequality [tex]\( -2x + 9 < x - 9 \)[/tex], we need to solve the inequality step by step. Here is a detailed solution:
1. Start with the given inequality:
[tex]\[
-2x + 9 < x - 9
\][/tex]
2. Move all terms involving [tex]\( x \)[/tex] to one side and constants to the other side:
[tex]\[
-2x + 9 - x < x - 9 - x
\][/tex]
Simplifying the equation, we get:
[tex]\[
-3x + 9 < -9
\][/tex]
3. Combine the constants on both sides of the inequality:
[tex]\[
-3x < -9 - 9
\][/tex]
Simplifying the right-hand side, we get:
[tex]\[
-3x < -18
\][/tex]
4. Now, divide both sides by -3. Remember, when you divide or multiply both sides of an inequality by a negative number, you must flip the inequality sign:
[tex]\[
x > 6
\][/tex]
5. Therefore, the solution set for the inequality [tex]\( -2x + 9 < x - 9 \)[/tex] is:
[tex]\[
x > 6
\][/tex]
6. On a number line, this solution means all numbers greater than 6. If you visualize it, you would have an open circle at 6 (indicating 6 is not included) and a shaded region extending to the right towards positive infinity.
So, the correct answer is the number line that shows an open circle at 6 with the shading extending to the right.
Clearly, this solution corresponds to the number line option, but since the options are not provided explicitly in text form, you should choose the one that has:
- An open circle at [tex]\( x = 6 \)[/tex].
- A shading to the right of 6 (indicating all numbers greater than 6).
Hence, the correct answer is the one that visually represents [tex]\( x > 6 \)[/tex].
