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Inequalities

Before we start, make sure we remember these important facts:

  • < means less than
  • > means greater than
  • ≤ means less than or equal to
  • ≥ means greater than or equal to
  • ≥ means greater than or equal to
  • Use an open circle for < or > when graphing
  • Use a closed circle for ≤ or ≥ when graphing

Equation vs. Inequality

There is only one difference between an equation and an inequality

An equation has an equal sign: =

An inequality has one of these: <, >, ≤, ≥

Example #1

Open Solve and Graph One-Step Inequalities in One Variable Example #1 in a new tab

Graphing Inequalities

Remember the following when graphing inequalities:

Use an open circle for < or >

Use a closed circle for ≤ or ≥

Example #2

Graph the solution for x < 8.

  1. Step One: Draw a number line
    Number line with the values 5, 6, 7, 8, 9, 10 listed
  2. Step Two: Will you use an open or closed circle?
    1. open circle
    2. closed circle

    Answer: a. open circle

  3. Step Three: Determine which way your arrow goes by substituting a number in for the variable to make the statement true. Then, draw the arrow pointing in the correct direction.
    Number line with the values 5, 6, 7, 8, 9, 10 listed with an open circle at 8 and a ray starting at 8 going to the left

Example #3

Open Solve and Graph One-Step Inequalities in One Variable Example #3 in a new tab

Example #4

Open Solve and Graph One-Step Inequalities in One Variable Example #4 in a new tab

Example #5

Open Solve and Graph Two-Step Inequalities in One Variable Example #5 in a new tab

Example #6

Graph the inequality x < 12.

Open circle or closed circle?

Which direction should the arrow go?

Graph the inequality.

Number line with the values 10, 11, 12, 13, 14 listed with an open circle at 12 and a ray starting at 12 going to the left

Example #7

Graph the inequality 42 ≥ m.

Open circle or closed circle?

Which direction should the arrow go?

Graph the inequality.

Number line with the values 40, 41, 42, 43, 44 listed with an open circle at 42 and a ray starting at 42 going to the left

Important Rule: Dividing or Multiplying By a Negative

When you divide or multiply by a negative value, you must flip the inequality sign of your final answer.

This does not apply to adding or subtracting.

Example #8

Solve and graph the inequality −4x ≤ 12.

First, divide both sides by −4 to isolate the variable. Because you are dividing by a negative value, you must remember to flip your sign.

3 lines. Line 1: negative 4 x is less than or equal to 12 original equation. Line 2: negative 4 x over negative 4 is less than or equal to 12 over negative 4. Line 3: x is greater than or equal to negative 3 division property.

Don't forget to flip the sign because you divided by a negative value.

Graph this inequality.

  1. number line with a closed circle at negative 3 with a line shaded to the left of the closed circle ending in an arrow to the left towards negative infinity
  2. number line with a closed circle at negative 3 with a line shaded to the right of the closed circle ending in an arrow to the right towards positive infinity
  3. number line with an open circle at negative 3 with a line shaded to the left of the open circle ending in an arrow to the left towards negative infinity
  4. number line with an open circle at negative 3 with a line shaded to the right of the open circle ending in an arrow to the right towards positive infinity

Answer:

number line with a closed circle at negative 3 with a line shaded to the right of the closed circle ending in an arrow to the right towards positive infinity

Graph B is correct. You have a closed point at −3 and all values "greater than or equal to" this point will make the inequality true.

Check your answer using the original inequality. Apply the Symmetric Property and substitute −3 in the original equation.

  • −4x = 12
  • −4(___blank) = 12
  • −4(−3) = 12
  • ___blank = 12
  • 12 = 12

When simplified, you can determine that your answer does check.

Example #9

Solve and graph the inequality:

negative 1 is greater than the fraction with numerator x plus 10 and denominator negative 4

Apply the symmetric property to rewrite the inequality with the variable on the left side.

the fraction with numerator x plus 10 and denominator negative 4 is less than negative 1

Apply the Multiplication Property and eliminate the denominator by multiplying on both sides. What else is important here? When multiplying by a negative value, the inequality sign must be flipped.

negative 4 times the fraction with numerator x plus 10 and denominator negative 4 is less than negative 1 times negative 4

The denominator will be eliminated on the left side, so simplify the right side.

3 lines. Line 1: the fraction with numerator x plus 10 and denominator negative 4 is less than negative 1. Line 2: negative 4 times the fraction with numerator x plus 10 and denominator negative 4 is less than negative 1 times negative 4. Line 3: x plus 10 is greater than 4.

Remember to flip your sign!

Isolate the variable on the left side of the inequality. Apply the Subtraction Property and subtract 10 on both sides of the inequality.

5 lines. Line 1: the fraction with numerator x plus 10 and denominator negative 4 is less than negative 1. Line 2: negative 4 times the fraction with numerator x plus 10 and denominator negative 4 is less than negative 1 times negative 4. Line 3: x plus 10 is greater than 4. Line 4: Subtract 10 from both sides of the equation. Line 5: x is greater than negative 6.

To graph the inequality x > −6, will you have an open or closed point?

  1. Open Point
  2. Closed Point

Answer: a. Open Point.

Will you shade your graph to the left or to the right of the point?

  1. Left
  2. Right

Answer: b. Right.

Your inequality is x > −6, so you will have an open point and your graph will be shaded to the right.

Choose the correct graph for the inequality x > −6.

  1. number line with an open point at negative 6 and a line shaded to the left of the open point that ends in an arrow pointing left towards negative infinity
  2. number line from negative 10 to negative 3. Closed point at negative 6 and a line shaded to the right of the closed point that ends in an arrow pointing right towards positive infinity
  3. number line with an open point at negative 6 and a line shaded to the right of the open point that ends in an arrow pointing right towards positive infinity
  4. number line from negative 10 to negative 3. Closed point at negative 6 and a line shaded to the left of the closed point that ends in an arrow pointing left towards negative infinity

Answer: Graph C.

number line with an open point at negative 6 and a line shaded to the right of the open point that ends in an arrow pointing right towards positive infinity

Check the answer x > −6 using the original inequality.

Apply the substitution property to substitute the answer into the original inequality. Change the inequality sign to an equal sign.

2 lines. Line 1: negative 1 equals the fraction with numerator x plus 10 and denominator negative 4. Line 2: negative 1 equals negative 6 over negative 4 plus 10.

Simplify −6 + 10 in the numerator. Next, you will simplify the fraction.

4 lines. Line 1: negative 1 equals the fraction with numerator x plus 10 and denominator negative 4. Line 2: negative 1 equals negative 6 over negative 4 plus 10. Line 3: negative 1 equals the fraction 4 over 4. Line 4: negative 1 equals negative 1.

You should find that your answer checks!