Question

On a certain marathon course, a runner reaches a big hill that is at least 10 miles into the race. If a total marathon is 26.2 miles, how can you find the number of miles the runner still has to go?

Answer 1

It looks like this problem involves inequality. Does that sound right?

Answer 2

I'll give you a hand. And I'll assume this involves inequality.

Since they say "at least 10 miles" into the race, how would you describe that [distance] as a number or potential numbers? You have to think about that.

So "at least 10" could be 10 at the least, but possibly greater. So, 10 or more. Like, 11, 12, and 13 would all work...
If you had to describe this length as a variable, like x, you would say that x is equal to 10 or greater. x=10 or x>10. Thus
\[x\ge10\]

Now, if you know how far you've gone [(it's x, in miles)], then you can find how much is less with subtraction. More exactly, 26.2 - x is the distance you have left to go - specifically the difference between how far you are in and how far the race goes.

26.2 - x = r, where r is the rest you have to go.
Then x = 26.2 - r.
So you can use substitution.
Don't do this: \[(\ge10) 26.2 - r\], because that makes no sense.

Do\[26.2 - r \ge10\]

Then solve.
given
\[26.2 - r \ge 10\]
add r to both sides\[26.2\ge10+r\]
and then subtract 10 from both sides\[16.2\ge r\]
and then flip it\[r\le 16.2\]

I hope this has helped!

When you add or subtract to all parts of an inequality, signs don't change. They all increase or decrease together. \[4<7\]\[4+1=5<8=7+1\]\[4-1=3<6=7-1\]

When you multiply or divide by a positive number, signs don't change. They all increase or decrease proportionally.\[4<7\]\[4\times2=8<14=7\times 2\]\[4\div 2 = 2< 3\frac{1}{2}=7\div2\]

When you multiply or divide by a positive number, signs flip direction, but anything that might be equal still might be equal. This is because you're now turning everything negative.\[4<7\]\[4\times -1 = -4 > -7 = 7\times -1\]\[4\div -2 = -2 > -3\frac{1}{2} = 7 \div -2\]

You could think -4 is still closer to 0 than -7 is, but we're talking negatives, so closer to 0 is actually a greater value. Multiplying or dividing by a negative is flipping all the signs.
When you look at a positive number, closer to 0 is lesser. When you look at a negative number, closer to 0 is now greater.