As elements go across from left to right in a period, they get closer to filling up their outermost electron energy shell. Their protons hold tightly onto their electrons. As they get closer to filling up their outermost electron energy shell, would it be harder or easier to take an electron away from them?

it would be harder to take an electron away.

Explanation:

The ionization energy is the energy required to remove an electron from the valence shell of an isolated gaseous atom of an element.

The atomic size decreases as we move from left to right in a period because the valence shell come closer to the nucleus as we move from left to right. Thus, the removal of electron from the valence shell becomes difficult due to greater attraction between nucleus and valence electron. Thus, the ionization energy increases from left to right in a period.

Thus it would be harder to take an electron away.