How to Do Two-Step Inequalities

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Equations are integral parts of mathematics. In order to succeed, a student must learn how to properly solve such things as 2x + 2 = 4. The number of actions used to solve the equation equals the number of "steps" used to solve it. When it comes to inequalities, the problems look similar. Instead of 2x + 2 = 4, the problem may switch the equal-sign to a greater-than, a less-than, or a greater/less-than-equal-to (< or > with a line beneath it) to show that the answer will be larger than the number given. Other than this change in signs, however, the solution process is virtually the same.

Things You'll Need

  • Pencil
  • Eraser
  • Scratch paper
  • Calculator (if desired)

Solving a Two-Step Inequality

  • Read the problem. Note whether the inequality is asking for a greater-than (>), less-than (<), or combination (> with a single line beneath: greater-than-equal-to, for example). All inequalities will ask for one of those four. Example: 6x - 7 > 15.

  • Write the inequality down on a piece of scratch paper. Check to make sure the problem has been written correctly. Make sure to allow for plenty of room to work out the problem. If necessary, erase with a good eraser and rewrite the problem.

  • Solve any parentheses first. Example: 3(2x + 7) < 60. In this example, multiply the 3 to the 2x and the 7 before doing anything else. The result is 6x + 21 < 60. Keep an eye out for any negatives that may alter the + or - signs used. Example: -3(-2x - 7) < 60 solves as 6x + 21 < 60 and not -6x - 21 < 60.

  • Add or subtract the non-variable numbers on both sides. Example: 2x + 7 < 60. Subtract the 7 from both sides of the inequality to end up with 2x < 53. The same applies for 2x - 7 < 60: the result would be 2x < 67.

  • Multiply or divide the variable on both sides to cancel them out. Example: 2x < 53. Divide by 2 on both sides. The result: x < 53/2, or x < 26.5. Note that multiplying or dividing by a negative number at this stage will reverse the inequality, regardless. Example: -2x < 53 will solve as x > -26.5.

  • Check your work by adding your answer for X back into the original problem. Example: 2x + 7 < 60, and the solution was x = 26.5.

    2(26.5) + 7 = 60

    53 + 7 = 60

    60 = 60

    Because 53 + 7 does equal 60, the answer is true and correct. Use a calculator if desired.

Tips & Warnings

  • Always make sure to watch your negatives and positives: they can change a number and will effect the outcome.
  • If the solution does not fit when put through the original problem, double-check your work carefully. It may have been a simple matter of saying 11 + 5 = 15 rather than 16.
  • Be careful of the inequality sign with negatives. Any time you multiply or divide the VARIABLE on both sides by a negative, the sign will flip. This does not include any multiplication or division that you do prior to getting rid of anything negative attached to the x (variable).

References

  • Photo Credit Stockbyte/Stockbyte/Getty Images
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