Question

Christy is taking a road trip. After she drives 12 more miles, she will have driven at least half of the 108-mile trip. What is the least number of miles she has driven so far?

Short answer

Christy will have driven at least 42 miles.

Step-by-step solution

Step 1. State the concept of linear inequalities.

Alinear inequalityis a mathematical statement that relates a linear expression as either less than or greater than another.

A solution to a linear inequality is a real number that will produce a true statement when substituted for the variable. Linear inequalities have either infinitely many solutions or no solution.

Step 2. Form the inequality from the given information. 

Assume the least number of miles driven by Christy to be ‘x’.

After driving 12 more miles, the minimum distance travelled by Christy is:$x+12$

After driving 12 more miles, Christy will have driven at least half of the 108-mile trip.

This means Christy will have driven more than or equal to half of the 108-mile trip.

Thus the required inequality is as follows:

$x+12\ge \frac{108}{2}\dots \left(1\right)$.

Step 3. Calculate the value of ‘x’.

Solve the inequality (1) to find the least number of miles driven by Christy.

$$\begin{gathered} x+12\ge \frac{108}{2} \\ x+12\ge 54 \end{gathered}$$

Add‘$-12$’ on both the sides of inequality.

$$\begin{gathered} x+12-12\ge 54-12 \\ x\ge 42 \end{gathered}$$

Thus, Christy will have driven at least 42 miles.