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Inequalities with Addition and Subtraction

Simplify one-step inequalities to identify solution set

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Solve Inequalities Using Addition or Subtraction
License: CC BY-NC 3.0

The graduating class of 2015 has been busy raising money to decorate the Community Center for the upcoming Prom Dance. They have a budget of $2,475. 95 and have already committed $1,728.50 to a professional decorator who will provide and set up the props to transform the back portion of the Community Center into a nature park. The graduates also want to have the tables and chairs covered with yellow, metallic cloth if they have enough money to cover the cost. They are still waiting to hear from “All Things Fancy” regarding the price. How can the graduates figure out how much money they have for covering the tables and chairs?

In this concept, you will learn to solve inequalities using addition or subtraction.

Adding and Subtracting Inequalities

Sometimes, you will have an inequality that is not as straightforward as \begin{align*}x>4\end{align*}. With this example, you know that the variable can be any number that is greater than four. The set of numbers that will make this inequality a true statement can be displayed as a list or graphed on a number line.

However, the set of numbers that makes the inequality true is not always as obvious as it was for the inequality \begin{align*}x>4\end{align*}. You may see an inequality like this one.

\begin{align*}x+3>7\end{align*}

For this inequality you need to find the set of numbers which will make the inequality a true statement. You are looking for a set of numbers such that the sum of a number and 3 is greater than 7. You could spend a lot of time doing trial and error to determine the set of numbers or you could solve the inequality.

Solving an inequality is similar to solving an equation. The following rules or number properties can be used to solve an inequality.

The Addition Property of Inequality states that if the same number is added to both sides of an inequality then the sense (inequality symbol) of the inequality remains unchanged.

  • If \begin{align*}a>b\end{align*}, then \begin{align*}a+c > b+c\end{align*}.
  • If \begin{align*}a < b\end{align*}, then \begin{align*}a+c < b+c\end{align*}.
  • If \begin{align*}a \ge b\end{align*}, then \begin{align*}a+c \ge b+c\end{align*}.
  • If \begin{align*}a \le b\end{align*}, then \begin{align*}a+c \le b+c\end{align*}.

The Subtraction Property of Inequality states that if the same number is subtracted from both sides on the inequality then the sense (equality symbol) of the inequality remains unchanged.

  • If \begin{align*}a>b\end{align*}, then \begin{align*}a-c > b-c\end{align*}.
  • If \begin{align*}a < b\end{align*}, then \begin{align*}a-c < b-c\end{align*}.
  • If \begin{align*}a \ge b\end{align*}, then \begin{align*}a-c \ge b-c\end{align*}.
  • If \begin{align*}a \le b\end{align*}, then \begin{align*}a-c \le b-c\end{align*}.

The inequalities which you will solve can be solved by applying these properties. The process involved in solving an equation is the same for solving an inequality – applying inverse operations to isolate and solve for the variable. For an equation the solution is a single number but the solution for an inequality will be a set of numbers. The solution must make the inequality a true statement.

Let’s look at solving an inequality.

Solve this inequality for all real numbers and graph the solution on a number line.

\begin{align*}n-3<5\end{align*}

First, isolate the variable by adding (inverse of subtracting) 3 to both sides of the inequality.

\begin{align*}\begin{array}{rcl} n-3 & < & 5\\ n-3+3 & < & 5+3 \end{array}\end{align*}

You have just applied the addition property of inequality. The inequality symbol remained the same.

Next, simplify both sides of the inequality.

\begin{align*}\begin{array}{rcl} n-3+3 & < & 5+3\\ n& < & 8 \end{array}\end{align*}

The solution set is \begin{align*}\{ n | n < 8, n \in R \}\end{align*}.

The solution is the set of numbers “less than” 8 and this can be graphed on a number line.

First, draw a number line from 0 to 10.

Next, place an open circle on the number 8 since this number is not included in the solution set.

Then, draw a direction line from 8 to the left on the number line.

License: CC BY-NC 3.0

Examples

Example 1

Earlier, you were given a problem about the graduating class of 2015. They need to know how much money they have left in their decorating budget.

They need to write an inequality to model their decorating funds.

First, write a verbal model to represent the information.

Verbal Model: cost of covering the tables and chairs + cost of the decorator cannot exceed their budget.

Next, write an equality to represent the verbal model.

\begin{align*}\underbrace{\text{cost of covering tables and chairs}}_{c} + \underbrace{\text{cost of decorator}}_{\$ 1,728.50} \le \underbrace{\text{total budget}}_{\$2,475.95}\end{align*}

The inequality is

\begin{align*}c+1,728.50 \le 2,475.95\end{align*} 

Next, solve the inequality for the variable “\begin{align*}c\end{align*}”.

First, isolate the variable by subtracting 1,728.50 from both sides of the inequality.

\begin{align*}\begin{array}{rcl} c+1,728.50 & \le & 2,475.95\\ c+1,728.50-1,728.50 & \le & 2,475.95 - 1,728.50 \end{array}\end{align*}

Then, simplify both sides of the inequality.

\begin{align*}\begin{array}{rcl} c+1,728.50-1,728.50 & \le & 2,475.95 - 1,728.50\\ c & \le & 747.45\\ \end{array}\end{align*}

The cost for covering the tables and chairs must be less than or equal to $747.45.

Example 2

Solve the following inequality for all real numbers. Then graph the solution on a number line.

\begin{align*}-2 \le x+4\end{align*}

First, isolate the variable by subtracting 4 from both sides of the inequality.

\begin{align*}\begin{array}{rcl} -2 & \le & x+4\\ -2-4 & \le & x+4-4 \end{array}\end{align*}

Next, simplify both sides of the inequality.

\begin{align*}\begin{array}{rcl} -2-4 & \le & x+4-4\\ -6 & \le & x\\ \end{array}\end{align*}

The solution is \begin{align*}\{ x | x \ge -6, x \in R \}\end{align*}.

If \begin{align*}-6 \le x\end{align*} then \begin{align*}x \ge -6\end{align*}. Regardless of how you write it, the solution remains the same.

Now, graph the solution on a number line.

First, draw an appropriate number line.

Next, place a closed circle on -6 since this value is included in the solution set.

Then, draw a direction line to the right to indicate “greater than” and to show the real number set.

Remember, joining the points means that all values between the whole numbers are included in the solution set.

License: CC BY-NC 3.0

Example 3

Solve the following inequality for the set of Integers.

\begin{align*}x+5 < -12\end{align*}

First, isolate the variable by subtracting 5 from both sides of the inequality.

\begin{align*}\begin{array}{rcl} x+5 & < & -12\\ x+5-5 & < & -12-5 \end{array}\end{align*} 

Next, simplify both sides of the inequality.

\begin{align*}\begin{array}{rcl} x+5-5 & < & -12-5\\ x & < & -17 \end{array}\end{align*}

Then, write the solution for the set of Integers.

\begin{align*}\{ x | x < -17, x \in I \}\end{align*}

Example 4

Solve the following inequality for all rational numbers.

\begin{align*}3+5x \ge 4x+6\end{align*}

Remember the variables must be on one side of the inequality sign and the numbers on the other side.

First, subtract 3 from both sides of the inequality.

\begin{align*}\begin{array}{rcl} 3+5x & \ge & 4x+6\\ 3-3+5x & \ge & 4x+6-3 \end{array}\end{align*}

Next, simplify both sides of the inequality.

\begin{align*}\begin{array}{rcl} 3-3+5x & \ge & 4x+6-3\\ 5x & \ge & 4x+3 \end{array}\end{align*}

Next, subtract \begin{align*}4x\end{align*} from sides of the inequality to get the variables on the left side of the inequality.

\begin{align*}\begin{array}{rcl} 5x & \ge & 4x+3\\ 5x-4x & \ge & 4x-4x+3 \end{array}\end{align*}

Next, simplify both sides of the inequality.

\begin{align*}\begin{array}{rcl} 5x-4x & \ge & 4x-4x+3\\ x & \ge & 3 \end{array}\end{align*}

Then, write the solution for the set of rational numbers.

\begin{align*}\{ x | x \le 3, x \in Q \}\end{align*}

Review

Solve the following inequalities:

  1. \begin{align*}x+4>10\end{align*}
  2. \begin{align*}y-11<20\end{align*}
  3. \begin{align*}a+2<1\end{align*}
  4. \begin{align*}b+3 \ge 5\end{align*}
  5. \begin{align*}y-2 \le -4\end{align*}
  6. \begin{align*}x+1 \ge 5\end{align*}
  7. \begin{align*}x-3 < 11\end{align*}
  8. \begin{align*}x-4 < -3\end{align*}
  9. \begin{align*}x-4>-3\end{align*}
  10. \begin{align*}y+7>22\end{align*}
  11. \begin{align*}a-6 \ge -1\end{align*}
  12. \begin{align*}b+14 > 20\end{align*}
  13. \begin{align*}x-24 > -11\end{align*}
  14. \begin{align*}a+3 \le -9\end{align*}
  15. \begin{align*}x+13 > -33\end{align*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 3.13. 

 

Vocabulary

Addition Property of Inequality

You can add a quantity to both sides of an inequality and it does not change the sense of the inequality. If x > 3, then x+2 > 3+2.

Equation

An equation is a mathematical sentence that describes two equal quantities. Equations contain equals signs.

inequality

An inequality is a mathematical statement that relates expressions that are not necessarily equal by using an inequality symbol. The inequality symbols are <, >, \le, \ge and \ne.

Subtraction Property of Inequality

The subtraction property of inequality states that the subtraction of equal amounts from both sides of an inequality will not change the sense of the inequality.

Image Attributions

  1. [1]^ License: CC BY-NC 3.0
  2. [2]^ License: CC BY-NC 3.0
  3. [3]^ License: CC BY-NC 3.0

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