The word problem in the image is about finding the possible dimensions of a garden whose area is less than 30 square meters, and where the length is 1 meter more than the width. Let's go through the problem-solving process shown in the images:

Given:

1. Area is less than 30 sq. meters.
2. Length is 1 meter more than the width.

Let $x$ be the width of the garden, and the length will be $x + 1$.

Inequality Setup:

We know the area of a rectangle is given by $\text{Length} \times \text{Width}$, which leads to the inequality: $(x + 1) \times x < 30$ This simplifies to the quadratic inequality: $x^2 + x < 30$ Rewriting it as an equation for solving: $x^2 + x = 30$ Move 30 to the left-hand side: $x^2 + x - 30 = 0$

Solving the Quadratic Equation:

The equation $x^2 + x - 30 = 0$ is factored as: $(x + 6)(x - 5) = 0$ The solutions for $x$ are: $x = -6 \quad \text{or} \quad x = 5$ Since width cannot be negative, we discard $x = -6$, leaving: $x = 5$

Testing the Range for Inequality:

Now, we need to test values to determine the valid range of widths.

From the second image:

• The critical points are $x = 0$ and $x = 5$, and testing shows that the inequality $x^2 + x < 30$ holds for $0 < x < 5$.

Conclusion:

The possible values for $x$ (width) are between 0 and 5 meters. Therefore, the possible dimensions of the garden (width by length) are:

• $1 \, m \times 2 \, m$
• $2 \, m \times 3 \, m$
• $3 \, m \times 4 \, m$
• $4 \, m \times 5 \, m$

Would you like further details or have any questions? Here are 5 related questions you might find useful:

1. How do you determine the critical points in a quadratic inequality?
2. Why is it necessary to test values in an inequality after solving the quadratic equation?
3. What happens if both roots of the quadratic equation are negative?
4. How do we interpret the solution set of $0 < x < 5$?
5. Can this approach be applied to other geometric shapes like triangles or circles?

Tip: Always test the boundary values in inequality problems to confirm if they are included in the solution set.