# Equations and Inequalities Exercises

#### Checking Solutions to Equations

How Do We Check if a Value is a Solution to an Equation?An equation works like this: (left-hand side expression) = (right-hand side expression). Therefore, an equation is only true if the left-hand...

#### Number of Solutions to an Equation

How Many Solutions Can an Equation Have?Okay, algebristas,* different equations can have different numbers of solutions. Right now, we are only counting solutions from the real numbers. Some equati...

#### Adding and Subtracting Constants

Sample ProblemConsider the equation x = 5. This equation claims that x is equal to 5.We can create equivalent equations to x = 5 by adding or subtracting the same amount of weight to...

#### Adding and Subtracting Variables

Just as we can add and subtract constants from both sides of an equation, we can also add and subtract copies of the variable from both sides of the equation. Therefore, if the same variable appear...

#### Multiplication and Division

Sample ProblemSolve the equation 8x = 5x + 24.The first thing we do is subtract 5x from each side to find that 3x = 24.We haven't talked yet about what to do with this sort of equation, but yo...

#### Simplifying Equations

Sometimes parentheses make equations look more complicated than they actually are. (See how much more complicated this sentence seems?) If you simplify the expressions on each side of the equation...

#### Eliminating Fractions

Aw...what did they ever do to you?When lots of fractions are involved, there is another way to make an equation look simpler before solving it: get rid of the fractions. Sweep them away, pack them...

#### Keeping Both Solutions

Equations of the form x2 = (some positive number) have two solutions: the positive and the negative square roots of that positive number. That's okay—there is room here for a second answer. The m...

#### When You Get Stuck

Like a truck. In the mud. Sometimes, when working out a math problem, we get stuck. Our tires are spinning like crazy, but we aren't moving forward a hair. Here's a neat trick: figuring out why...

#### Solving Equations with Multiple Variables

When an equation has more than one variable, we can't just say "solve the equation." That's like telling you to "name the One Tenor." You can't do that, because there are three of them. Not that ma...

#### Strict Inequalities

These inequalities don't take no guff. They lay down the law. Better not cross 'em.The symbol "" abbreviates "is greater than." We personally think the latter symbol is superior, which we can expre...

#### Equivalent Inequalities

Equivalent inequalities are inequalities with the same solutions. They are also the most likely to get paired up together on eHarmony. It's incredible how much they have in common. It's like they w...

#### Inequalities that Allow Equality

Okay, time out. How can you be an inequality and allow equality? Isn't that an oxymoron? Can you also perform an injustice that is just? Or have indigestion while digesting?The answer is that some...

#### Solving Inequalities

When we solved equations, we found all values of the variable that made the equation true. Similarly, we can solve inequalities to find all values of the variable that make the inequality true....