Question

1. maria has a rectangular patio in her backyard and the area of the patio is no more than 45 square meters. what are the values of x for which the area of the rectangle will be no more than 45 square meters? hint: $a = l(w)$

inequality: ____________
solution: $x\geq6 + 20$ (make sure it is an inequality and labeled!!)
(the rectangle has side lengths $x + 3$ and $4.5$ m)

Explanation:

Step1: Set up area inequality

The area of a rectangle is $A = L \times W$. Here, $L = 4.5$, $W = x+3$, and $A \leq 45$.
$$4.5(x+3) \leq 45$$

Step2: Divide both sides by 4.5

Isolate the term with $x$.
$$x+3 \leq \frac{45}{4.5}$$
$$x+3 \leq 10$$

Step3: Subtract 3 from both sides

Solve for $x$, and note $x+3>0$ (width can't be non-positive, so $x>-3$; this is automatically satisfied since $x\leq7$ and $x>-3$ is a broader condition here).
$$x \leq 10 - 3$$
$$x \leq 7$$
Also, since the width $x+3$ must be positive:
$$x+3 > 0 \implies x > -3$$
Combining with the area condition, the valid range is $-3 < x \leq 7$, but since $x$ represents a length-related value, $x$ must be non-negative in context, so $0 \leq x \leq 7$.

Answer:

Inequality: $\boldsymbol{4.5(x+3) \leq 45}$ (simplified to $\boldsymbol{x \leq 7}$ with $x \geq 0$ for practical context)