Solving and Graphing One- and Two-Step Inequalities - Seventh Grade Mathematics

Linear inequalities define continuous ranges of mathematical solutions, governed by inverse operations and the essential sign-reversal rule for negative multipliers.

Inequality Symbols and Solution Sets

An inequality compares two expressions that are not necessarily equal, establishing that one side is greater than, less than, or equal to the other.

+-----------------------+-----------------------------+------------------------+
| Symbol                | Meaning                     | Number Line Circle     |
+-----------------------+-----------------------------+------------------------+
| <                     | Strictly less than          | Open circle (O)        |
| >                     | Strictly greater than       | Open circle (O)        |
| <=                    | Less than or equal to       | Closed / solid dot (*) |
| >=                    | Greater than or equal to    | Closed / solid dot (*) |
+-----------------------+-----------------------------+------------------------+

Unlike an equation, which typically has a single unique numeric solution, an inequality has an infinite set of solutions. The statement x is greater than three includes three point one, four, one hundred, and an infinity of values extending to the right.

Graphing Inequalities on a Number Line

To graph an inequality on a number line:
1. Plot a boundary circle at the numerical solution value.

2. Use an open circle for strict inequalities (< or >) to show the boundary value itself is excluded.

3. Use a closed solid dot for inclusive inequalities (<= or >=) to show the boundary value is included in the solution set.

4. Shade the number line in the direction that satisfies the inequality (to the right for greater than, to the left for less than).

Graph of x > 2:
  <----+----+----+----+----+----+----+----+----+---->
      -1    0    1    2    3    4    5    6    7
                      O========================>  (Open circle, shaded right)

Graph of x <= -1:
  <----+----+----+----+----+----+----+----+----+---->
      -4   -3   -2   -1    0    1    2    3    4
  <===================*                           (Solid dot, shaded left)

The Golden Rule: Reversing the Inequality Sign

When solving inequalities, you apply the exact same inverse operations used for equations, with one vital, non-negotiable exception: whenever you multiply or divide both sides by a negative number, you MUST reverse the direction of the inequality sign.

               THE NEGATIVE REVERSAL PRINCIPLE
   If you multiply or divide an inequality by a NEGATIVE number:
      <  becomes  >              >  becomes  <
      <= becomes  >=             >= becomes  <=

To understand why this mathematical reversal is mandatory, examine a simple true statement on a number line: two is less than five (2 < 5).

Start with true statement:        2  <  5
Multiply both sides by -1:       -2  ?  -5
On a number line, -2 is to the RIGHT of -5:
                                 -2  >  -5   (The sign must flip!)

Multiplying by a negative reflects values across the origin zero to the opposite side of the number line. Values that were larger on the positive side become further to the left (smaller) on the negative side. Failing to reverse the inequality sign turns a true mathematical relationship into a false one.

Solving Two-Step Inequalities

Solving a two-step inequality requires undoing addition or subtraction first, then undoing multiplication or division.

Example: Solve -4x + 7 >= 31

Step 1 (Subtract 7 from both sides):
         -4x + 7 - 7 >= 31 - 7
         -4x >= 24

Step 2 (Divide both sides by -4 and FLIP the sign):
         -4x / (-4) <= 24 / (-4)
         x <= -6

The solution set is all real numbers less than or equal to negative six. On a number line, place a closed solid dot at negative six and shade the arrow pointing left.

Check a test point: Let x = -10 (which is <= -6):
  -4(-10) + 7 = 40 + 7 = 47
  Is 47 >= 31? Yes, true statement!

Testing a point from your shaded region confirms your solution.

Exercises

Work through each problem carefully in your notebook before reading the provided solutions.

Exercise 1. Solve the one-step inequality negative three x is greater than or equal to twenty-four, and describe how to graph its solution set on a number line.

Exercise 2. Solve the two-step inequality five x minus eight is strictly less than twenty-seven.

Exercise 3. Solve the two-step inequality negative two-thirds y plus four is greater than ten.

Exercise 4. A student solves the inequality negative six x minus five is less than or equal to nineteen and concludes that x is less than or equal to negative four. Identify the student's mathematical error and state the correct solution.

Solutions

Solution to Exercise 1. In the inequality negative three x is greater than or equal to twenty-four, divide both sides by negative three. Because you are dividing by a negative number, the greater than or equal to sign must reverse to less than or equal to: x is less than or equal to twenty-four divided by negative three, which equals negative eight. The solution is x is less than or equal to negative eight. To graph this on a number line, draw a closed solid dot at negative eight and shade the arrow extending indefinitely to the left.

Solution to Exercise 2. The inequality is five x minus eight is less than twenty-seven. First undo subtraction by adding eight to both sides: twenty-seven plus eight equals thirty-five, yielding five x is less than thirty-five. Next divide both sides by positive five. Since five is positive, the inequality sign does not change: x is strictly less than seven. To graph, place an open circle at positive seven and shade to the left.

Solution to Exercise 3. The inequality is negative two-thirds y plus four is greater than ten. First subtract four from both sides: ten minus four equals six, yielding negative two-thirds y is greater than six. Next multiply both sides by the reciprocal of negative two-thirds, which is negative three-halves. Because you are multiplying by a negative number, reverse the greater than sign to a less than sign: y is less than six times negative three-halves. Six divided by two is three, and three times negative three is negative nine. The solution is y is strictly less than negative nine.

Solution to Exercise 4. The student forgot to reverse the inequality sign when dividing by negative six. The correct steps are: add five to both sides of negative six x minus five is less than or equal to nineteen: nineteen plus five equals twenty-four, giving negative six x is less than or equal to twenty-four. Now divide both sides by negative six and reverse the inequality sign from less than or equal to greater than or equal to: x is greater than or equal to twenty-four divided by negative six, which equals negative four. The correct solution is x is greater than or equal to negative four.