To solve the inequality [tex]\(-7x > 21\)[/tex], we need to follow a few steps. Here's a detailed, step-by-step solution:
1. Understand the inequality:
We are given [tex]\(-7x > 21\)[/tex]. To isolate [tex]\(x\)[/tex], we need to get rid of the coefficient [tex]\(-7\)[/tex] that is multiplied by [tex]\(x\)[/tex].
2. Divide both sides by [tex]\(-7\)[/tex]:
When we divide both sides of an inequality by a negative number, we must reverse the direction of the inequality sign.
[tex]\[
\frac{-7x}{-7} < \frac{21}{-7}
\][/tex]
This simplifies to:
[tex]\[
x < -3
\][/tex]
3. Interpret the result:
The inequality [tex]\(x < -3\)[/tex] means that [tex]\(x\)[/tex] can be any value less than [tex]\(-3\)[/tex].
When graphing the solution of this inequality on a number line:
- We make an open circle (or hollow dot) at [tex]\(-3\)[/tex] to show that [tex]\(-3\)[/tex] is not included in the solution.
- We then shade the number line to the left of [tex]\(-3\)[/tex] to represent all values less than [tex]\(-3\)[/tex].
4. Graph of the solution:
The graph of [tex]\(x < -3\)[/tex] will look like this:
[tex]\[
\begin{array}{c}
\ \ \ \ \ \ \ \ \| \ \ \ \ \ \ \ \ \ \| \ \ \ \ \ \ \ \ \|\ \ \ \ \ \ \ \\
\----------------------------------------------------------------\\
<\ -------->\ \ \ \ \ \ -3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ( = omitted \ \
\end{array}
\][/tex]
This graphical representation shows all the values to the left of [tex]\(-3\)[/tex] as part of the solution.
