Answer :
To simplify the expression \(|2x + 7|\), we need to consider the properties of the absolute value function. The absolute value \(|u|\) of a function \(u\) is defined as:
[tex]\[ |u| = \begin{cases} u & \text{if } u \geq 0 \\ -u & \text{if } u < 0 \end{cases} \][/tex]
In this case, \(u = 2x + 7\). We need to determine the break point where \(2x + 7 = 0\), as this divides the cases to consider. Solving for \(x\):
[tex]\[ 2x + 7 = 0 \\ 2x = -7 \\ x = -\frac{7}{2} \][/tex]
So,
- If \(x > -\frac{7}{2}\), \(2x + 7 > 0\).
- If \(x = -\frac{7}{2}\), \(2x + 7 = 0\).
- If \(x < -\frac{7}{2}\), \(2x + 7 < 0\).
Now, we can write the absolute value expression in its different cases:
1. If \(x > -\frac{7}{2}\):
- In this region, \(2x + 7\) is positive, so \(|2x + 7| = 2x + 7\).
2. If \(x = -\frac{7}{2}\):
- Here, \(2x + 7 = 0\), so \(|2x + 7| = 0\).
3. If \(x < -\frac{7}{2}\):
- In this region, \(2x + 7\) is negative, so \(|2x + 7| = -(2x + 7)\).
So the expression for \(|2x + 7|\) fully simplified considering all cases is:
[tex]\[ |2x + 7| = \begin{cases} 2x + 7 & \text{if } x > -\frac{7}{2} \\ 0 & \text{if } x = -\frac{7}{2} \\ -(2x + 7) = -2x - 7 & \text{if } x < -\frac{7}{2} \end{cases} \][/tex]
Therefore, filling in the blanks:
[tex]\[ \begin{matrix} \text{If } x > & -\frac{7}{2} & \text{, then } |2x + 7| = & 2x + 7 \\ \text{If } x = & -\frac{7}{2} & \text{, then } |2x + 7| = & 0 \\ \text{If } x < & -\frac{7}{2} & \text{, then } |2x + 7| = & -2x - 7 \end{matrix} \][/tex]
[tex]\[ |u| = \begin{cases} u & \text{if } u \geq 0 \\ -u & \text{if } u < 0 \end{cases} \][/tex]
In this case, \(u = 2x + 7\). We need to determine the break point where \(2x + 7 = 0\), as this divides the cases to consider. Solving for \(x\):
[tex]\[ 2x + 7 = 0 \\ 2x = -7 \\ x = -\frac{7}{2} \][/tex]
So,
- If \(x > -\frac{7}{2}\), \(2x + 7 > 0\).
- If \(x = -\frac{7}{2}\), \(2x + 7 = 0\).
- If \(x < -\frac{7}{2}\), \(2x + 7 < 0\).
Now, we can write the absolute value expression in its different cases:
1. If \(x > -\frac{7}{2}\):
- In this region, \(2x + 7\) is positive, so \(|2x + 7| = 2x + 7\).
2. If \(x = -\frac{7}{2}\):
- Here, \(2x + 7 = 0\), so \(|2x + 7| = 0\).
3. If \(x < -\frac{7}{2}\):
- In this region, \(2x + 7\) is negative, so \(|2x + 7| = -(2x + 7)\).
So the expression for \(|2x + 7|\) fully simplified considering all cases is:
[tex]\[ |2x + 7| = \begin{cases} 2x + 7 & \text{if } x > -\frac{7}{2} \\ 0 & \text{if } x = -\frac{7}{2} \\ -(2x + 7) = -2x - 7 & \text{if } x < -\frac{7}{2} \end{cases} \][/tex]
Therefore, filling in the blanks:
[tex]\[ \begin{matrix} \text{If } x > & -\frac{7}{2} & \text{, then } |2x + 7| = & 2x + 7 \\ \text{If } x = & -\frac{7}{2} & \text{, then } |2x + 7| = & 0 \\ \text{If } x < & -\frac{7}{2} & \text{, then } |2x + 7| = & -2x - 7 \end{matrix} \][/tex]
