Question $-3(3 p-1) \leq 7 p+51$

Studdy Solution

STEP 1

What is this asking? We need to find all values of $p$ that make this inequality true! Watch out! Flipping the inequality sign when multiplying or dividing by a negative number is super important!

STEP 2

1. Distribute and Simplify
2. Isolate the Variable
3. Final Inequality

STEP 3

Let's distribute that $-3$ on the left side.
Remember, we're multiplying $-3$ by both terms inside the parentheses.
This gives us: $-3 \cdot 3p + (-3) \cdot (-1) \leq 7p + 51$ $-9p + 3 \leq 7p + 51$ So, we've simplified the left side of the inequality!

STEP 4

Now, let's get all the $p$ terms on one side.
We can subtract $7p$ from both sides: $-9p + 3 - 7p \leq 7p + 51 - 7p$ $-16p + 3 \leq 51$ Look at that, the $p$ terms are all on the left!

STEP 5

Next, let's subtract $3$ from both sides to isolate the $p$ term: $-16p + 3 - 3 \leq 51 - 3$ $-16p \leq 48$ Almost there!

STEP 6

Finally, we divide both sides by $-16$ to solve for $p$ .
Remember to flip the inequality sign since we're dividing by a negative number: $\frac{-16p}{-16} \geq \frac{48}{-16}$ $p \geq -3$ Boom! We found our solution!

STEP 7

Our final answer is $p \geq -3$ .
This means any value of $p$ that is greater than or equal to $-3$ will make the original inequality true!

STEP 8

$p \geq -3$

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