To determine the solution to the inequality [tex]\( 15 - 5x < -10 \)[/tex], follow these steps:
1. Isolate the variable term: First, we need to move the constant term on the left-hand side of the inequality over to the right-hand side. We do this by subtracting 15 from both sides.
[tex]\[
15 - 5x - 15 < -10 - 15
\][/tex]
Simplifying this, we get:
[tex]\[
-5x < -25
\][/tex]
2. Solve for [tex]\( x \)[/tex]: To isolate [tex]\( x \)[/tex], we need to divide both sides of the inequality by [tex]\(-5\)[/tex]. Remember that dividing or multiplying both sides of an inequality by a negative number reverses the inequality sign.
[tex]\[
\frac{-5x}{-5} > \frac{-25}{-5}
\][/tex]
Simplifying this, we get:
[tex]\[
x > 5
\][/tex]
3. Interpret the solution: The inequality [tex]\( x > 5 \)[/tex] means that [tex]\( x \)[/tex] can be any number greater than 5.
4. Represent the solution on the number line: To represent [tex]\( x > 5 \)[/tex] on a number line:
- Draw a number line with numbers marked on it.
- Place an open circle (not filled in) at 5 to indicate that 5 is not included in the solution.
- Shade the number line to the right of 5 to indicate all numbers greater than 5 are included in the solution.
The number line representation of the solution [tex]\( x > 5 \)[/tex] looks like this:
- A number line with points labeled.
- An open circle at 5.
- Shading or an arrow extending to the right (positive side) beyond the 5.
In summary, the solution to the inequality [tex]\( 15 - 5x < -10 \)[/tex] is all values of [tex]\( x \)[/tex] greater than 5, and this is represented on the number line with an open circle at 5 and shading to the right.
