130 Part 2: Into the Unknown
CHECK POINT
Solve each equation. If there is no solution, say that. If all numbers are solutions, label
the equation an identity.
16. 2x – 3 = x – 3- 6x – 4 = 2(3x – 2)
- 9x + 11 = 3(3x + 4)
19. 5x + 13 = 22 – 4x
20. 19 – 8x = 4(5 – 2x) – 1
One Solution or Many?
When you solve an equation, you find the value of the variable that makes the two sides of the
equation identical. It’s the one number that makes the statement true. There are more compli-
cated equations for which there is more than one solution. We’re not ready to look at the solution
of those equations just yet, but it’s good to know they exist. For example, the equation x^2 = 9 has
two solutions. If you replace x with 3, you have 3^2 = 9, which is true, but if you replace x with -3,
you get (-3)^2 = 9, which is also true. The equation x^2 = 9 has two solutions.
There’s another group of problems that are true for more than one value of the variable, and
these are a type of problem we are ready to solve. They’re called inequalities because rather than
saying “this equals that,” they say, “this is bigger than that,” or “this is smaller than that.” They
tell us that the two sides are unequal.DEFINITION
An inequality is a mathematical sentence that says one expression is greater than
another.The symbol > is read “is greater than,” and the symbol < is read “is less than.” If you add a line
under the symbol, you add “or equal to.” So x≤ 4 says “x is less than or equal to 4,” and y≤^3 -
says “y is less than or equal to -3. If you think about those statements for a minute, you can see
that there are many numbers that you could put in place of x to make a true statement. You could
say 5 u 4 or 19 u 4 and both would be true. There are many, many more substitutions for x that
make true statements—an infinite number of them, in fact.
Equations ask you to find the value for the variable that makes both sides the same. Inequalities
ask you to find the values that make one side larger than the other. The solution will be a set of
numbers, rather than a single value.