Why does your home central air conditioner run but not blow cold air?

do you have your a/c set to fan? not a/c? otherwise you might need to contact your local a/c repair man and e will tell you you need a charge of freon in your system.

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Why not to start checking the simple points like the filter, duct mechanism, etc.

Not so simple to give a "One Size Fits All" answer for but here are a few things to check.

1) when the system is running does the outside unit kick on? That is the condensing unit and supports the compressor which ultimately provides the cooling to the coil in the unit inside.

2) is the air filter dirty or do you have low air flow if so check the lines at the outside unit see if they're frosting or icing up. If they are try the air filter but if that still doesn't solve the problem call for service a low freon charge will cause icing.

3) if you have low flow but a clean filter check the "MERV" rating on the filter a merv 6-8 should be sufficient for most applications except cigarette smoke. The lower the merv # the less the resistance to air flow the larger the particle rating that it will filter in microns.

4) There are other components which can require servicing but they are normally handled by your service person unless you are familiar with HVAC repair. Have a Great day.