Question

1. what is the solution to \\(\frac{3}{2}b + 5 < 17\\)? 1) \\(b < 8\\) 2) \\(b > 8\\) 3) \\(b < 18\\) 4) \\(b > 18\\)

Explanation:

Step1: Subtract 5 from both sides

To isolate the term with \( b \), we subtract 5 from both sides of the inequality \(\frac{3}{2}b + 5 < 17\). This gives us \(\frac{3}{2}b + 5 - 5 < 17 - 5\), which simplifies to \(\frac{3}{2}b < 12\).

Step2: Multiply both sides by \(\frac{2}{3}\)

To solve for \( b \), we multiply both sides of the inequality \(\frac{3}{2}b < 12\) by the reciprocal of \(\frac{3}{2}\), which is \(\frac{2}{3}\). So we have \(\frac{2}{3} \times \frac{3}{2}b < 12 \times \frac{2}{3}\). The left side simplifies to \( b \) and the right side simplifies to \( 8 \). So we get \( b < 8 \). Wait, no, wait. Wait, \( 12\times\frac{2}{3}=\frac{24}{3} = 8\)? Wait, no, wait, let's recalculate. Wait, \( 12\times\frac{2}{3}\): 12 divided by 3 is 4, 4 times 2 is 8? Wait, no, that's wrong. Wait, \(\frac{3}{2}b < 12\), multiply both sides by \(\frac{2}{3}\): \( b < 12\times\frac{2}{3}\). 12 times 2 is 24, 24 divided by 3 is 8? Wait, no, that would be \( b < 8 \), but let's check again. Wait, the original inequality is \(\frac{3}{2}b + 5 < 17\). Subtract 5: \(\frac{3}{2}b < 12\). Then multiply both sides by \(\frac{2}{3}\): \( b < 12\times\frac{2}{3}\). 12 times 2 is 24, 24 divided by 3 is 8? Wait, no, 12 divided by 3 is 4, 4 times 2 is 8. So \( b < 8 \)? But wait, let's test with \( b = 8 \): \(\frac{3}{2}(8)+5 = 12 + 5 = 17\), but the inequality is less than 17, so \( b < 8 \) would make \(\frac{3}{2}b + 5 < 17\). Wait, but let's check the options. Option 1 is \( b < 8 \), option 3 is \( b < 18 \). Wait, maybe I made a mistake. Wait, no, let's do it again. \(\frac{3}{2}b + 5 < 17\). Subtract 5: \(\frac{3}{2}b < 12\). Multiply both sides by \(\frac{2}{3}\): \( b < 12\times\frac{2}{3}\). 12 times 2 is 24, 24 divided by 3 is 8. So \( b < 8 \). So the correct option is 1) \( b < 8 \). Wait, but let's check with \( b = 7 \): \(\frac{3}{2}(7)+5 = \frac{21}{2}+5 = 10.5 + 5 = 15.5 < 17\), which works. If \( b = 8 \): \(\frac{3}{2}(8)+5 = 12 + 5 = 17\), which is not less than 17. If \( b = 9 \): \(\frac{3}{2}(9)+5 = 13.5 + 5 = 18.5 > 17\), so the inequality holds when \( b < 8 \). So the correct answer is option 1.

Wait, but earlier I thought I made a mistake, but no, the calculation is correct. So the solution is \( b < 8 \), which is option 1.

Answer:

1. \( b < 8 \)