Certainly! Let's work through the problem step-by-step to derive the correct inequality used to find the possible lengths of the base of the triangle.
1. Understanding the problem:
- The height [tex]\( h \)[/tex] of the triangle is 4 inches greater than twice its base [tex]\( x \)[/tex].
- The area of the triangle is no more than 168 square inches.
2. Expressing the height in terms of the base:
- Let [tex]\( x \)[/tex] be the base of the triangle.
- Then the height [tex]\( h \)[/tex] can be expressed as:
[tex]\[
h = 2x + 4
\][/tex]
3. Formula for the area of the triangle:
- The area [tex]\( A \)[/tex] of a triangle is given by:
[tex]\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\][/tex]
- Substituting the expressions for base and height, we get:
[tex]\[
A = \frac{1}{2} \times x \times (2x + 4)
\][/tex]
4. Given that the area is no more than 168 square inches:
- This translates to the inequality:
[tex]\[
\frac{1}{2} \times x \times (2x + 4) \leq 168
\][/tex]
5. Simplifying the inequality:
- First, multiply both sides of the inequality by 2 to get rid of the fraction:
[tex]\[
x \times (2x + 4) \leq 336
\][/tex]
- Distribute [tex]\( x \)[/tex] inside the parenthesis:
[tex]\[
2x^2 + 4x \leq 336
\][/tex]
- Simplify by dividing all terms by 2:
[tex]\[
x^2 + 2x \leq 168
\][/tex]
6. Restating the inequality in standard form:
- This final form expresses a quadratic inequality:
[tex]\[
x (x + 2) \leq 168
\][/tex]
Thus, the correct 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]
So the answer is:
[tex]\[
\boxed{x(x+2) \leq 168}
\][/tex]
