Question

1. the solution to $4p + 2 < 2(p + 5)$ is
2. $p > -6$
3. $p < -6$
4. $p > 4$
5. $p < 4$

Explanation:

Step1: Expand the right side

We start with the inequality \(4p + 2 < 2(p + 5)\). First, we expand the right - hand side using the distributive property \(a(b + c)=ab+ac\). Here, \(a = 2\), \(b=p\) and \(c = 5\), so \(2(p + 5)=2p+10\). The inequality becomes \(4p + 2<2p + 10\).

Step2: Subtract \(2p\) from both sides

To get all the \(p\) terms on one side, we subtract \(2p\) from both sides of the inequality. \((4p+2)-2p<(2p + 10)-2p\). Simplifying the left - hand side: \(4p-2p + 2=2p+2\), and the right - hand side: \(2p-2p + 10 = 10\). So the inequality is now \(2p+2<10\).

Step3: Subtract 2 from both sides

Next, we subtract 2 from both sides to isolate the term with \(p\). \((2p + 2)-2<10 - 2\). The left - hand side simplifies to \(2p\), and the right - hand side simplifies to \(8\). So we have \(2p<8\)? Wait, no, wait. Wait, in step 2, we had \(4p + 2<2p+10\). Let's re - do step 2 correctly. Subtract \(2p\) from both sides: \(4p-2p+2<2p - 2p+10\), which is \(2p+2<10\). Then subtract 2 from both sides: \(2p+2 - 2<10 - 2\), so \(2p<8\)? No, wait, the original inequality is \(4p + 2<2(p + 5)\). Let's expand \(2(p + 5)\) correctly: \(2\times p+2\times5 = 2p + 10\). So the inequality is \(4p+2<2p + 10\). Now, subtract \(2p\) from both sides: \(4p-2p+2<2p-2p + 10\), so \(2p+2<10\). Then subtract 2 from both sides: \(2p+2 - 2<10 - 2\), so \(2p<8\). Then divide both sides by 2: \(p < 4\). Wait, but let's check again. Wait, maybe I made a mistake. Let's start over.

Starting with \(4p+2<2(p + 5)\)

Step1: Expand the right - hand side

\(2(p + 5)=2p+10\), so the inequality is \(4p + 2<2p+10\)

Step2: Subtract \(2p\) from both sides

\(4p-2p+2<2p-2p + 10\)
\(2p+2<10\)

Step3: Subtract 2 from both sides

\(2p+2 - 2<10 - 2\)
\(2p<8\)

Step4: Divide both sides by 2

\(\frac{2p}{2}<\frac{8}{2}\)
\(p < 4\)

Answer:

1. \(p < 4\)