Question

How do you write a compound inequality to represent the scenario. You'll need to bring at least \[\$15\] to the movies but you won't need more than \[\$25\]. Let m represent the money brought to the movies?

Answer 1

Hint: The information gives the minimum and the maximum amount of money which will be needed.
"At least \[\$15\]" means that amount or more than that.
"At most or not more than \[\$25\]" means that amount or less.
In both cases the given value is included.
In question they mentioned that m represents the money brought to the movies.
\[\$\] m represents the money brought to the movies.

Complete step by step solution:
Now we need to write the compound inequality to represent the scenario that money brought to the movies should be at least \[\$15\] but it should not be more than \[\$25\].
In the first case,
They mentioned in the question that the money brought to the movies should be at least \[\$15\].
We know that \[\$\] m represents the money brought to the movies.
Now the inequality of the first case becomes:
\[\$\] m should be at least \[\$15\].
\[\$\] m should be greater than or equal to \[\$15\]. It should not be less than \[\$15\]. So,
\[\Rightarrow \$ \] m \[\ge \] \[\$15\].
In the second case,
They mentioned in the question that the money brought to the movies should not be more than \[\$25\].
We know that \[\$\] m represents the money brought to the movies.
Now the inequality of the second case becomes:
\[\$\] m should not be more than \[\$25\].
\[\$\] m should be less than or equal to \[\$25\]. It should not be greater than \[\$25\].So,
\[\Rightarrow \$ \] m \[\le \] \[\$25\].
Now combining both the cases we get the compound inequality of the money brought to the movies.
First case is:
\[\Rightarrow \$ \] m \[\ge \] \[\$15\].
Second case is:
\[\Rightarrow \$ \] m \[\le \] \[\$25\].
From both the cases, the amount of money is \[\$\] m and the compound inequality becomes:
\[\Rightarrow \] \[\$15\] \[\le \$ \] m \[\le \] \[\$25\]
This is the math way of saying “any amount from \[\$15\] to \[\$25\]”.
Hence, the compound inequality to represent the scenario that money brought to the movies should be at least \[\$15\] but it should not be more than \[\$25\] is:
\[\Rightarrow \$15 \] \[\le \] m \[\le \] \[\$25\]

Note: Students must know the concept of inequality expressions. Students must know the concept of compound inequality. Compound inequality is an inequality that combines two simple equations. To solve the compound inequality first separate it into two inequalities.