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# Equations and Inequalities

# Solutions to Equations

### What is a Solution to an Equation?

Let's start with the basics: An **equation** is a string of mathematical symbols stating the equality of two expressions.

### Sample Problems

1. *x* = 5 means "*x* equals 5."

2. "*x* times two plus seven equals three times *x*" can be written in symbols as 2*x* + 7 = 3*x*.

3. The equation states that the quantities and 10 are equal.

An equation is a claim that two quantities are equal. It's like if someone's all, "Hey man, flammable is the same as inflammable," and you're like, "no way, prove it." Think of that equal sign as "is," so when *x* = *y* - 5, we are saying that *x* *is* *y* - 5. An equation is true if those two quantities actually are equal, and it is not true ("false") if those two quantities are unequal. In our example of the gentleman who seems to be concerned about a fire hazard, we would write "flammable = inflammable." If these two terms are exactly equal, then the equation is true.

As it turns out, they *are* the exact same thing. Probably good to know.

### Sample Problems

1. The equation 3 + 1 = 2 + 2 is true, since 3 + 1 and 2 + 2 are each 4.

2. The equation 1 = 0 is false because 1 and 0 are not the same value.

When an equation contains a variable, the equation may be true for some values of that variable and false for others. A **solution** to an equation is a value that, when substituted for the variable, makes the equation true. Here, we could say "a piece of fruit drawn at random (variable) out of this box is a banana." If some fruits in the box are bananas and some are not, then this statement may sometimes be true and sometimes not. Hopefully true in this case, since potassium is an important part of a balanced diet.

### Sample Problems

4 is a solution to the equation 2*x* = *x* + 4, since 2(4) does equal (4) + 4.

3 is not a solution to the equation 2*x* = *x* + 4, since 2(3) = 6 but 3 + 4 = 7.