Let's find the correct inequality to express the possible lengths, [tex]\( x \)[/tex], of the base of the triangle given the conditions provided.
1. Define the Variables:
- Let [tex]\( x \)[/tex] be the base of the triangle (in inches).
- The height of the triangle is 4 inches greater than twice its base, which can be expressed as [tex]\( 2x + 4 \)[/tex].
2. Recall the Formula for the Area of a Triangle:
- The area [tex]\( A \)[/tex] of a triangle is given by [tex]\( A = \frac{1}{2} \times \text{base} \times \text{height} \)[/tex].
3. Substitute the Given Values into the Area Formula:
- Substitute [tex]\( x \)[/tex] for the base and [tex]\( 2x + 4 \)[/tex] for the height:
[tex]\[
A = \frac{1}{2} \times x \times (2x + 4)
\][/tex]
4. Set up the Inequality Based on the Given Condition:
- It’s given that the area of the triangle is no more than 168 square inches. Thus, we have the inequality:
[tex]\[
\frac{1}{2} \times x \times (2x + 4) \leq 168
\][/tex]
5. Simplify the Inequality:
- First, simplify the expression inside the inequality:
[tex]\[
\frac{1}{2} \times x \times (2x + 4) \leq 168
\][/tex]
- Simplify the multiplication:
[tex]\[
\frac{1}{2} \times (2x^2 + 4x) \leq 168
\][/tex]
- Distribute the [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[
x^2 + 2x \leq 168
\][/tex]
6. Express the Final Inequality:
- The resulting inequality is:
[tex]\[
x^2 + 2x \leq 168
\][/tex]
- This can be rewritten as:
[tex]\[
x(x + 2) \leq 168
\][/tex]
Thus, the inequality that can be used to find the possible lengths [tex]\( x \)[/tex] of the base of the triangle is:
[tex]\[
x(x + 2) \leq 168
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{x(x+2) \leq 168}
\][/tex]
