To solve the compound inequality [tex]\(18 \leq 2c + 6 < 32\)[/tex], we need to break it down into two separate inequalities and then solve each one individually. After solving, we will combine the results to find the range of possible values for [tex]\( c \)[/tex].
Step 1: Solve the first inequality [tex]\(18 \leq 2c + 6\)[/tex]
Subtract 6 from both sides to isolate the term with [tex]\(c\)[/tex]:
[tex]\[ 18 - 6 \leq 2c \][/tex]
[tex]\[ 12 \leq 2c \][/tex]
Next, divide both sides by 2 to solve for [tex]\(c\)[/tex]:
[tex]\[ \frac{12}{2} \leq \frac{2c}{2} \][/tex]
[tex]\[ 6 \leq c \][/tex]
So, the solution to the first inequality is:
[tex]\[ c \geq 6 \][/tex]
Step 2: Solve the second inequality [tex]\(2c + 6 < 32\)[/tex]
Subtract 6 from both sides to isolate the term with [tex]\(c\)[/tex]:
[tex]\[ 2c + 6 - 6 < 32 - 6 \][/tex]
[tex]\[ 2c < 26 \][/tex]
Next, divide both sides by 2 to solve for [tex]\(c\)[/tex]:
[tex]\[ \frac{2c}{2} < \frac{26}{2} \][/tex]
[tex]\[ c < 13 \][/tex]
So, the solution to the second inequality is:
[tex]\[ c < 13 \][/tex]
Step 3: Combine the results
We need to find the values of [tex]\(c\)[/tex] that satisfy both inequalities simultaneously. From the first inequality, we have [tex]\( c \geq 6 \)[/tex], and from the second inequality, we have [tex]\( c < 13 \)[/tex].
Therefore, the range of [tex]\(c\)[/tex] that satisfies both conditions is:
[tex]\[ 6 \leq c < 13 \][/tex]
To express this in interval notation, we write:
[tex]\[ c \in [6, 13) \][/tex]
This means [tex]\( c \)[/tex] can take any value between 6 and 13, including 6 but not including 13.
