1. **Problem Statement:** We have two rational numbers $a$ and $b$ on a number line, where $a < 0 < b$. The number $c$ is defined as the sum $c = a + b$. We need to determine which of the following statements are true: $c < a$, $c < b$, $|c| < |a|$, $|c| < |b|$. 2. **Understanding the problem:** Since $a$ is negative and $b$ is positive, adding them means $c$ could be negative, zero, or positive depending on their magnitudes. 3. **Analyze each statement:** - $c < a$ means $a + b < a$ which simplifies to $b < 0$. But $b$ is positive, so this is **false**. - $c < b$ means $a + b < b$ which simplifies to $a < 0$. Since $a$ is negative, this is **true**. - $|c| < |a|$ means the absolute value of $a + b$ is less than the absolute value of $a$. Since $a$ is negative, $|a| = -a$. Whether this is true depends on the relative sizes of $a$ and $b$. For example, if $b$ is small, $|c|$ could be less than $|a|$. But if $b$ is large, $|c|$ could be greater. So this is **not always true**. - $|c| < |b|$ means the absolute value of $a + b$ is less than the absolute value of $b$. Since $b$ is positive, $|b| = b$. Similar to the previous case, this depends on the relative sizes of $a$ and $b$. So this is **not always true**. 4. **Summary:** - $c < a$: False - $c < b$: True - $|c| < |a|$: Not always true - $|c| < |b|$: Not always true **Final answer:** Only $c < b$ is always true.