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Solving Two-Step Inequalities with Fractions

Instructor: Maria Airth

Maria has a Doctorate of Education and over 15 years of experience teaching psychology and math related courses at the university level.

Solving two-step inequalities is very similar to solving two-step equations. This lesson covers the steps and gives examples and practice for solving two-step inequalities with fractions.

Inequalities

An inequality is simply a mathematical sentence that is not balanced. Think of a balance scale; an equation would have each side of the scale perfectly balanced, whereas an inequality results in one side of the scale being higher than the other. In an equation, the sides are separated by an equal sign (=). In an inequality, the two sides of the sentence are separated by an inequality sign, which indicates which side is larger through reference to the first side (> greater than or < less than). For example, 11 < 1/2x + 1 indicates that 11 is less than the value of 1/2x + 1.

Add an equal sign to the inequality symbol or just a line under it to indicate that the sides may be equal ( 11 <= x is read 11 is less than or equal to x). Inequalities that include an equal component work in exactly the same way as pure inequalities.

Two-Step Inequalities With Fractions

The specification of a two-step inequality means exactly what it sounds like; it takes two steps to solve the inequality. We can, again, use our knowledge of a two-step equation to help us here. In a two-step equation, we needed to do two things in order to isolate the variable.

  • Start: 11 = 1/2x + 1
  • First step: 11-1 = 1/2x + 1 -1; 10=1/2x
  • Second step: 10(2) = 1/2x(2) ; 20 = x

You can see the first step removes the constant from the side with the variable, and the second step completely isolates the variable to solve the equation. Inequalities work the same.

First Step- Add The Inverse

The first step in solving an inequality is to add the inverse of the constant. Remember to add it to both sides! Adding the inverse is just doing the opposite of what you see. Examples of inverses: '10 and -10' and '-5 and 5'. This requirement doesn't change if there is a fractional constant in the inequality; you still add the inverse of the fraction to both sides (removing it from the side without changing the ratio between the sides).

Let's look at some examples.

Example 1: 12 < 2x - 1/3. The first step is to add 1/3 to both sides of the equation to get 12 + 1/3< 2x or 37/3 < 2x.

Example 2: 3/5 > -3x + 2. Subtract 2 (add the inverse of 2 is the same as subtract 2) from both sides to get 3/5 - 2 > -3x or -7/5 > -3x.

Second Step - Multiply The Inverse

The second step in the process is to completely isolate the variable by multiplying both sides of the inequality by the inverse of the coefficient (the number attached to the variable). Multiplying by the inverse on both sides cancels out the coefficient without changing the ratio of values on the sides.

Let's go back to those earlier examples and start from where we left off.

Example 1: 37/3 < 2x. To multiply by the inverse, we need to multiply both sides by 1/2 (thus cancelling out the coefficient because 2 * 1/2 =1). We get 37/3 * 1/2 < x or 37/6 < x.

Example 2: -7/5 > -3x. Multiply both sides by -1/3 to get -7/5 * -1/3 <x or 7/15 < x.

Did you notice what happened to the inequality sign in example 2? When you multiply by a negative number on both sides of the inequality, you must flip the sign! Remember, with inequalities, if you multiply by a negative, flip the direction of the sign.

Optional Step (One)

When dealing with fractions, sometimes it is easier to deal with the fractions first and then move on to the problem. Do this by multiplying each term by the LCM (lowest common multiple) of the terms. This effectively turns all the fractions into whole numbers without changing the ratio of values from one side to the other.

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