From 11:00PM PDT on Friday, July 1 until 5:00AM PDT on Saturday, July 2, the Shmoop engineering elves will be making tweaks and improvements to the site. That means Shmoop will be unavailable for use during that time. Thanks for your patience!
We have changed our privacy policy. In addition, we use cookies on our website for various purposes. By continuing on our website, you consent to our use of cookies. You can learn about our practices by reading our privacy policy.
© 2016 Shmoop University, Inc. All rights reserved.

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-han...

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're only counting solutions from the real numbers. Some equat...

Adding and Subtracting Constants

Sample ProblemConsider the equation x = 5. This equation claims that x is equal to 5. That means the "weight"  of x is the same as the "weight" of 5, so let's get visual and draw a...

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 appe...

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, bu...

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 equat...

Eliminating Fractions

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

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's room here for a second answer. The...

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...

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 ex...

Equivalent Inequalities

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

Non-Strict Inequalities

Inequalities that Allow EqualityOkay, 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...

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....
Advertisement