Is the 7.5 you mention is a TSH test result? If so, were any other tests done? With the suspicion of hypothyroidism, if the 7.5 is a TSH test, then the doctor should have also tested for the possibility of Hashimoto's Thyroiditis. Two tests are required: Thyroid Peroxidase antibodies (TPO ab) and Thyroglobulin antibodies (TG ab). In addition, since TSH does not correlate well with the actual thyroid hormones, you should make sure they always test you for both Free T4 and Free T3. Note that these are not the same as Total T4 and Total T3, so make sure they are testing for the Frees. Most of the thyroid hormone in your blood is bound to protein and thus biologically inactive. The small portions not bound to protein are called Free T4 and Free T3. Those are what you need to know about.
If you are hypothyroid, then you also need to be tested for Vitamin D, B12 and ferritin (a storage form of iron). D should be at least 50 ng/mL, B12 in the upper end of its range, and ferritin should be at least 100. So you need to discuss this with the doctor and get tested and then supplement as needed. Do you think you can get the above mentioned 7 tests done?
What thyroid med was prescribed for you and what dosage?
There is more we need to discuss, but let's start with this.