Why doesn't Israel just shoot down the Hamas missiles?

Instead of invading Gaza, why doesn't Israel simply decide to shoot the Hamas missles out of the air?

Perhaps Israel could fire US made Raytheon Patriot missiles at the Hamas rockets.

It occurs to me that the answer probably comes down to a simple cost-benefit ratio. Patriot missiles are expensive (one estimate is $2 million per missile). Most Hamas missiles are not accurate and seldom hit targets of value. Moreover, the Hamas rockets that have been fired at Israel number in the thousands (see this chart).

I saw that contributors at RedIt -- a tech-savvy group -- have also attempted to answer this question. I have grouped their thoughts by topic:

A new anti-missile system is in the works

• I agree that they attacked Gaza because they do not yet have this capability. But I am looking into the future where they will not need to do so.
• Iron Dome (a new anti-missile system) is nearing completion. In fact, I expect a test in the next couple of months.
• Israel has had the system in development for a long time (research US-Israel missile joint development programs and you will find it) however it will not be ready for another 12 - 18 months.
• Iron Dome is ineffective against the rockets. This was well known by the developers and the Defense Ministry before development even began. It is a relatively worthless system and a major scam.

It's not worth it to Israel

• If Israel intercepted them, they wouldn't have an excuse to blow hundreds of Palestinians to bits.
• Hamas hardly ever manages to kill anyone with their rockets (NOT MISSILES)

It's impossible

• You can't shoot down the small, lo-tech missles that Hamas fires.
• There's no such thing as a "missile shield."
• Patriot missiles have a 100% failure rate.
• Hamas' homemade rockets are not sophisticated enough to be shot down by Israel's fancy US equipment.
• Those Hamas missiles were usually made with thin galvanized pipes and the whole contraptions is mobile & costing less than a hundred bucks.... then Israelis were going to use missiles costing at least USD50,000 a pop to shoot those toothpicks down?

• Q: Are you saying Israel is shredding people because someone flicked toothpicks at them?
  
• A: Yes that's exactly what is happening.
• Q: What's the cost of a bomb that can reduce an apartment building to rubble?
  
• A: to Israel? Zero dollars. Uncle sam pays for it.
  
• Q: How much does it cost to send a jet on a sortie?
  
• A: Same as above.

The question posed by this post touches on a deeper defense issue facing Israel. Highly destructive but relatively low-tech devices are becoming less expensive to install - whether it's roadside bombs in Iraq or rockets in Gaza. Hamas is using very low tech weapons, yet Israel is forced either to employee high-tech weapons to counter the threat, or launch full-scale military operations. It is expensive for Israel to destroy installations of cheap offensive weapons. Meanwhile, the global defense industry continues to convince the leaders of rich countries that they can invent or spend their way out of this conundrum. And Israel might win this battle, and maybe the next, but the one after that? For how long will Israel's advantage last -- and at what cost?

From the perspective of Israel, peacemaking should look like a bargain.

Chart: Israeli injuries (grey) and deaths (red) caused by Hamas-fired missiles. (via Muqata)