Inequalities: Worded problems

To solve the problem, we need to analyze the inequality involving the pages written by Writers A and B:

First, let $x$ represent the number of pages Writer B writes per day. Then, the number of pages Writer A writes is given by the expression $\frac{3}{5}x$.

The problem states that together they write more than 200 pages per day. Therefore, we can set up the inequality:

We need to simplify and solve this inequality:

• Combine the terms on the left-hand side:

Substituting this back into the inequality:

To solve for $x$, multiply both sides of the inequality by $\frac{5}{8}$ to isolate $x$:

Perform the multiplication:

This implies that the number of pages $x$ written by Writer B should be greater than 125.

Substitute $x > 125$ back to find the pages written by Writer A:

Therefore, Writer A writes more than 75 pages per day.

The correct answer is:

More than -75

To solve this problem, we will follow these steps:

• Step 1: Write the equation based on the problem statement.

• Step 2: Simplify and solve the inequality.

• Step 3: Identify the range for Simon's age.

Now, let's work through each step:
Step 1: The problem states that Gabriel is 14 years older than Simon, so if Simon's age is $S$, Gabriel's age is $S + 14$. Given that the sum of their ages does not exceed 35, we can write: $(S) + (S + 14) \leq 35$

Step 2: Simplify the inequality:
$\begin{aligned} 2S + 14 &\leq 35 \\ 2S &\leq 35 - 14 \\ 2S &\leq 21 \end{aligned}$ Divide both sides by 2:
$S \leq 10.5$

Step 3: Given the inequality, Simon’s age is any number greater than or equal to 0 but less than or equal to 10.5. Since Simon must be a whole number, Simon's possible ages range from 0 to 10.

After reviewing the given choices, the correct answer falls within the range:
Between 0 and 10.5

To solve this problem, follow these steps:

• Step 1: Identify that $x = \frac{1}{3} y$ and substitute into the inequality $x + y < 700$.
• Step 2: Rewrite the inequality as $\frac{1}{3} y + y < 700$.
• Step 3: Combine and simplify terms: $\frac{4}{3} y < 700$.
• Step 4: Solve for $y$ by multiplying both sides by $\frac{3}{4}$: $y < 525$.
• Step 5: Substitute back to find $x$: Since $x = \frac{1}{3} y$, $x < \frac{1}{3} \times 525 = 175$.

Thus, we conclude that the production of factory A, $x$, is less than 175 cartons per day.

To solve this problem, we'll contemplate the mathematical relationships between the sweets.

• Step 1: Expression Setup
  Mariano's sweets can be expressed as $5x + 4$, and Iván's sweets are $8x - 14$.
• Step 2: Inequality Setup
  Given that Iván has fewer sweets, establish an inequality: $8x - 14 < 5x + 4$.
• Step 3: Solve the Inequality
  Simplify the inequality:
  $8x - 14 < 5x + 4$
  Subtract $5x$ from both sides:
  $3x - 14 < 4$
  Add 14 to both sides:
  $3x < 18$
  Divide by 3:
  $x < 6$
• Step 4: Considering Logical Constraints
  Since sweets represent physical goods, $x$ must be positive. Therefore, $x > 0$.
• Step 5: Conclusion
  Combining these inequalities, $0 < x < 6$.

Therefore, the possible number of sweets that Daniel has is $0 < x < 6$.