Extended Display use old iMac

You cannot easily, Target Display Mode will not work.

iMac models introduced in 2011, 2012, 2013, and mid 2014

To use any of these iMac models as an external display:

• The iMac used as a display must have macOS High Sierra or earlier installed.
• The other Mac that you're connecting it to must have been introduced in 2019 or earlier and have macOS Catalina or earlier installed.
• The cable connecting the two Mac computers must be a Thunderbolt or Thunderbolt 2 cable.

24-inch and 27-inch iMac models introduced in 2009 and 2010

To use any of these iMac models as an external display:

• The iMac used as a display must have macOS High Sierra or earlier installed.
• The other Mac that you're connecting it to must have been introduced in 2019 or earlier and have macOS Catalina or earlier installed.
• The cable connecting the two Mac computers must be a Mini DisplayPort cable.

To get the same functionality that target display mode offers with Mac to Mac (and even get PC to Mac and iPad to Mac) support, Luna Display and Duet Display are two of the top options.

• Luna Display does require a $79 Luna Display USB-C dongle but has no subscription cost.
• Duet Display doesn’t require any hardware but goes from $25/year for Duet Air (to get Mac to Mac, PC to Mac, and iPad to Mac support). Duet Display offers a free trial.

https://9to5mac.com/2024/02/02/use-imac-as-monitor/

Sorry, but neither iMac will work with Apple's Target Display Mode.

Late 2014 & later iMac models do NOT support Target Display Mode. The other Mac that you're connecting it to must have been introduced in 2019 or earlier and have macOS Catalina or earlier installed.

Remember iMacs are computers, they are not external displays for other devices. Please do yourself a huge favor and just buy an external display.

Why?

No expensive cables needed

Less expensive

Dead Reliable

It sounds as if you purchased a 24" M3 iMac.

That Mac has two USB-C (USB, DIsplayPort, Thunderbolt 3) ports. It can drive one external display with up to 6K resolution at 60 Hz.

Most modern displays have resolutions of UHD 4K (3840x2160 pixels) or less – and support two (or more) of the following types of video inputs.

• DisplayPort – Use a USB-C to DisplayPort adapter or adapter cable.
• Mini DisplayPort – Use a USB-C to Mini DisplayPort adapter or adapter cable.
• HDMI – Use a USB-C to HDMI adapter or adapter cable.
• USB-C (DisplayPort) – Use a high-quality USB-C to USB-C cable. (If the monitor in question has downstream USB hub ports or Ethernet ports, a single USB-C cable will usually hook up both video and downstream ports.)

Some displays – mostly ones with 5K or 6K resolution – may require Thunderbolt input.

I would suggest looking for a display that has

• An IPS panel
• Near-100% coverage of sRGB. (Some people also look for near-100% coverage of wide gamuts like Adobe RGB or DCI-P3.). Most monitor vendors other than Apple specify the sRGB coverage, so if a monitor's specifications don't mention sRGB, assume the worst.
• Two or more modern inputs ((Mini) DisplayPort, HDMI, USB-C (DisplayPort)), preferably of two (or more) types. That increases your odds of being able to reuse the monitor with the next Mac you get after this one.

There are hundreds that would work beginning as low as $100.00. Visit Amazon to begin looking.

Yes, that should work, though several people complain bout the speed being slow.