Question 130·Medium·Algebra

"8 greater than $-2$ times the value of $q$" means you start with $-2q$ and then add 8, so that part should look like $-2q + 8$, not $8(-2q)$ and not $-2q - 8$.

After you write the inequality $3p \ge -2q + 8$, substitute $q = -5$ and simplify the right-hand side. You should get an inequality of the form $3p \ge$ (a number). Then divide both sides by 3 and think: what value of $p$ just barely satisfies this $\ge$ inequality?

In a new expression line, type (-2 * (-5) + 8) / 3 to represent $\dfrac{-2(-5) + 8}{3}$. The numerical result that Desmos shows is the least possible value of $p$ that satisfies the original inequality.

"Three times a number $p$" becomes $3p$.

"$-2$ times the value of $q$" becomes $-2q$.

"8 greater than $-2$ times the value of $q$" means $-2q + 8$ (you add 8 to $-2q$).

"Is at least" means "greater than or equal to," written as $\ge$.

So the sentence translates to the inequality

This relates $p$ and $q$. The next step is to use the given value of $q$.

We are told that $q = -5$. Substitute $-5$ for $q$ in the inequality:

Now simplify the right-hand side:

• $-2(-5) = 10$
• $10 + 8 = 18$

So the inequality becomes

Now solve this inequality for $p$.

To isolate $p$ in $3p \ge 18$, divide both sides by 3. Since 3 is positive, the inequality direction stays the same:

This means $p$ can be 6 or any number greater than 6. The least possible value that still satisfies the inequality is when $p$ equals the boundary value, so the least possible value of $p$ is 6.