To solve the inequality [tex]\( -3x - 7 < 20 \)[/tex] and determine which graph represents the solution, we need to isolate [tex]\( x \)[/tex] step by step. Here is the detailed approach:
1. Start with the given inequality:
[tex]\[
-3x - 7 < 20
\][/tex]
2. Add 7 to both sides to begin isolating [tex]\( x \)[/tex]:
[tex]\[
-3x - 7 + 7 < 20 + 7
\][/tex]
Simplify the equation:
[tex]\[
-3x < 27
\][/tex]
3. Divide both sides by [tex]\(-3\)[/tex] to solve for [tex]\( x \)[/tex]. Remember, when you divide or multiply by a negative number, the inequality sign reverses.
[tex]\[
\frac{-3x}{-3} > \frac{27}{-3}
\][/tex]
Simplify the equation:
[tex]\[
x > -9
\][/tex]
4. Conclusion: The solution to the inequality [tex]\( -3x - 7 < 20 \)[/tex] is [tex]\( x > -9 \)[/tex].
Graphically:
- You plot the number line.
- Mark the point [tex]\( x = -9 \)[/tex] on the number line.
- Use an open circle at [tex]\( x = -9 \)[/tex] (since [tex]\(-9\)[/tex] is not included in the solution, the inequality is strict).
- Shade the number line to the right of [tex]\(-9\)[/tex] to indicate all [tex]\( x \)[/tex] values greater than [tex]\(-9\)[/tex].
The graph that represents this solution should have an open circle at [tex]\( -9 \)[/tex] and a shaded region extending to the right towards positive infinity.
