Using the $C$% confidence interval for $\mu_1 - \mu_2$ to perform two sample $t$ test (equal variances not assumed)

Be sure you understand the theory behind confidence intervals and significance tests, before trying to understand the explanation behind the 'explain' button.
Two sided test If 0 is not in the $C$% confidence interval, the difference between the two sample means $\bar{y}_1 - \bar{y}_2$ is significantly different from 0 at significance level $\alpha = 1 - C/100$.
For instance, if 0 is not in the $95$% confidence interval, the difference between the two sample means $\bar{y}_1 - \bar{y}_2$ is significantly different from 0 at significance level $\alpha = 1 - 95/100 = .05$.
Explain
Right sided test If 0 is not in the $C$% confidence interval and the difference between the two sample means $\bar{y}_1 - \bar{y}_2$ is larger than 0, the difference between the two sample means $\bar{y}_1 - \bar{y}_2$ is significantly larger than 0 at significance level $\alpha = \frac{1 \, - \, C/100}{2}$.
For instance, if 0 is not in the $90$% confidence interval and the difference between the two sample means $\bar{y}_1 - \bar{y}_2$ is larger than 0, the difference between the two sample means $\bar{y}_1 - \bar{y}_2$ is significantly larger than 0 at significance level $\alpha = \frac{1 \,-\, 90/100}{2} = .05$.
Explain
Left sided test If 0 is not in the $C$% confidence interval and the difference between the two sample means $\bar{y}_1 - \bar{y}_2$ is smaller than 0, the difference between the two sample means $\bar{y}_1 - \bar{y}_2$ is significantly smaller than 0 at significance level $\alpha = \frac{1 \, - \, C/100}{2}$.
For instance, if 0 is not in the $90$% confidence interval and the difference between the two sample means $\bar{y}_1 - \bar{y}_2$ is smaller than 0, the difference between the two sample means $\bar{y}_1 - \bar{y}_2$ is significantly smaller than 0 at significance level $\alpha = \frac{1 \,-\, 90/100}{2} = .05$.
Explain