Let's solve the given problem step by step. We know that point [tex]\( P \)[/tex] is on line segment [tex]\( \overline{OQ} \)[/tex]. The problem provides the following expressions for the lengths of the segments:
- [tex]\( PQ = x + 7 \)[/tex]
- [tex]\( OP = 4x - 10 \)[/tex]
- [tex]\( OQ = 4x \)[/tex]
Since [tex]\( P \)[/tex] lies on the line segment [tex]\( \overline{OQ} \)[/tex], the sum of the lengths of [tex]\( \overline{OP} \)[/tex] and [tex]\( \overline{PQ} \)[/tex] must equal the length of [tex]\( \overline{OQ} \)[/tex]:
[tex]\[ OP + PQ = OQ \][/tex]
Substituting the given expressions into this equation, we get:
[tex]\[ (4x - 10) + (x + 7) = 4x \][/tex]
Now combine like terms on the left side of the equation:
[tex]\[ 4x - 10 + x + 7 = 4x \][/tex]
This simplifies to:
[tex]\[ 5x - 3 = 4x \][/tex]
Next, we solve for [tex]\( x \)[/tex] by isolating [tex]\( x \)[/tex] on one side of the equation:
[tex]\[ 5x - 4x = 3 \][/tex]
[tex]\[ x = 3 \][/tex]
Now that we have the value of [tex]\( x \)[/tex], we substitute [tex]\( x = 3 \)[/tex] back into the expression for the length of [tex]\( \overline{OQ} \)[/tex]:
[tex]\[ OQ = 4x \][/tex]
[tex]\[ OQ = 4(3) \][/tex]
[tex]\[ OQ = 12 \][/tex]
Thus, the numerical length of [tex]\( \overline{OQ} \)[/tex] is:
[tex]\[ OQ = 12 \][/tex]
