A major tool that the Federal Reserve has to try and control inflation is the Fed's influence on interest rates.
I would guess that the primary effect that rising interest rates has is to cool down asset prices. Inflation in existing home values; for instance. Homes are an asset. Home value/price inflation has been rampant for decades.
As time goes on, the rising cost of living creates more pressure to increase wages so people can still afford to live in an area. I would guess the Fed has less direct control over wages and most other prices in the economy, but inflating asset prices is the lever it can use to try and control the rest of inflation.
Asset price inflation also relates to stock prices and various other things that can add inflationary pressure to the economy. If interest rates are too low, corporations are (I would guess) more likely to add to their capital costs by buying out one another. Corporate mergers based on easy money.
With the big issue of housing, raising the interest rates can have both a positive "anti inflation" effect, but it can also have a negative, "more inflation" effect.
High interest rates can reduce the supply of housing by putting a damper on new construction. Eventually housing supply falls behind population growth in a region which can drive up prices.
So even this tool of cooling the overheated housing market is a blunt tool. It can have the opposite of it's intended purpose.
Since long time homeowners may be more apt to vote, inflation that was confined to assets was less noticed by politicians in decades past. This during the Bush, Obama, Trump years of easy money and continuing government deficits.
Now that wages have been going up, since Covid, inflation is more rampant in things like the cost of a restaurant meal or food at the grocery store. Inflation is now more "across the board."
The alarm bells are now ringing about inflation.
Quite a few other factors contribute to inflation as well, such as supply chain issues, wages, gas prices and so forth. These other factors may be less directly influenced by interest rates than inflation of assets, but it's all interconnected so it's all effected to some extent.
Gas prices are a factor, but they are influenced by a lot of things, such as the war between Russia and Ukraine. Rising interest rates may have less effect here.
Gasoline demand keeps going up while the future of oil production is in question, due to climate change.
There are a lot of moving parts in an economy. Many influences and not easy to pin blame on any one factor.
That's my take for today, I guess.
Sometimes I wake up in the morning and just start thinking about economics.
I recently read an article about the struggle that the Fed is having bringing down growth in wages since the job market is so strong these days. They are trying to at least cool wage growth a bit.
No mention of housing in that article, but I think the bigger influence, that the Fed has, would not be about wages, but about cooling the housing and asset markets. Wages is more along for the ride.