A question about IPTG induction
What makes you think that the lac repressor will still be bound at the target gene? 

In strains with this type of regulatory configuration (usually with DE3 in the genotype) the idea is that IPTG induction turns on expression of T7 polymerase from a lac promoter on the chromosome, and then this T7 polymerase then transcribes the target gene on the plasmid. The presence of IPTG ensures that repressor will bind at neither site.

Supplementary:

I think that I understand why you are confused now. Your picture of the way this works is based on the idea that the repressor is locked on to the operator until an inducer such as IPTG comes along. In fact there is an equilibrium between bound and unbound:


  repressor/operator &lt;=> repressor + operator


and, in the brief periods when the repressor isn't bound, the E. coli RNA polymerase can bind and transcribe the operon. This is referred to as 'escape synthesis' and it is generally assumed that it is functionally relevant in ensuring that there are low levels of lac permease expression at all times so that lactose can be taken up. If this wasn't the case, how would the natural inducer (a derivative of lactose) ever be able to induce the lac operon?

In protein expression systems it is generally considered desirable to keep escape synthesis of the foreign protein to a minimum so that there is no selection acting on the gene to negate any deleterious effects of its expression. The use of T7 polymerase as an intermediate level of regulation helps to achieve this. Even if there is a low level of escape synthesis of T7 polymerase, the concentration of T7 polymerase will be so low in the uninduced state (compared to the endogenous RNA polymerase) that the amount of escape synthesis of the target protein due to T7 polymerase gaining productive access to the T7 promoter on the plasmid will be negligible.

Read more here.