Try It

Compound Inequalities Matching

Practice Problem #1

Match the inequality to its description.

Descriptions:

  1. All numbers less than 5 and greater than −2
  2. All numbers greater than 5 or less than −2
  3. All numbers less than 5 or greater than −2
  4. All numbers less than −2 and greater than 5

Inequalities:

  • x > 5 or x < −2
  • x < 5 or x > −2
  • −2 < x < 5
  • x < −2 and x > 5

Answers:

  1. All numbers less than 5 and greater than −2

    Answer: −2 < x < 5

  2. All numbers greater than 5 or less than −2

    Answer: x > 5 or x < −2

  3. All numbers less than 5 or greater than −2

    Answer: x < 5 or x > −2

  4. All numbers less than −2 and greater than 5

    Answer: x < −2 and x > 5

Practice Problem #2

−1 ≤ x < 4

Match the phrases to the correct blanks in the following explanation.

Fill in the Blanks:

  1. Write the two phrases separately. All real numbers that are _______blank and all real numbers that are _______blank.
  2. Write the inequality that represents each statement.
    • x ≥ ___blank
    • x < ___blank
    • x ≥ −1 ___blank x < 4

Word Bank:

  1. −1
  2. less than 4
  3. 4
  4. greater than or equal to −1
  5. and

Answers:

    1. Write the two phrases separately. All real numbers that are greater than or equal to −1 and all real numbers that are less than 4.
    2. Write the inequality that represents each statement.
      • x ≥ −1
      • x < 4
      • x ≥ −1 and x < 4

Compound Inequalities Multiple Choice

Practice Problem #3

  1. Choose the graph that represents x ≥ −1 and x < 4.
    1. a number line from negative 2 to 5 with an open dot on negative 1 and a closed dot on 4 and a shaded line connecting the two points
    2. a number line from negative 2 to 5 with a closed dot on negative 1 and an open dot on 4 and a shaded line connecting the two points
    3. a number line from negative 2 to 5 with a closed dot on negative 1 and a shaded arrow going to the left and an open dot on 4 and a shaded arrow going to the right

    Answer: b.

    a number line from negative 2 to 5 with a closed dot on negative 1 and an open dot on 4 and a shaded line connecting the two points

Compound Inequalities Guided Practice

Practice Problem #4

Let's solve a compound inequality involving AND.

Graph the inequality −2 < 3y − 4 < 14.

  • −2 < 3y − 4 < 14
  • __blank < 3y − 4 and __blank < 14   Write the two inequalities separately
  • −2 < 3y − 4 and 3y − 4 < 14
  • 2 + 4 < 3y − 4 + 4 and 3y − 4 + 4 < 14 + 4   Add 4 to each side of the inequalities to isolate the y variable
  • __blank < 3y and __blank < __blank   Simplify each side
  • 6 < 3y and 3y < 18
  • fraction 6 over 3 less than fraction 3y over 3 and fraction 3y over 3 less than fraction 18 over 3
    Divide each inequality by 3 to isolate the y variable.
  • __blank < y and y < __blank   Simplify
  • 2 < y and y < 6
  • __blank < y < __blank   Rewrite as a single equality
  • 2 < y < 6
  • Now let's graph our compound inequality 2 < y < 6 on the number line.
    blank number line from negative 8 to 8
  • Answer:
    number line from negative 8 to 8 with an open circle at 2 and an open circle at 6 and shaded line connecting the two points

Practice Problem #5

Let's solve a compound inequality involving OR.

Graph the solutions for the compound inequality: −2y + 7 < 1 or 4y + 3 ≤ −5

  • First, break the inequality into two pieces
    • −2y + 7 < 1 or 4y + 3 ≤ −5
  • In the left hand inequality (−2y + 7 < 1), subtract 7 from both sides to isolate the y variable.
    • −2y + 7 − 7 < 1 − 7
  • In the right side inequality (4y + 3 ≤ −5), subtract 3 from both sides to isolate the y variable.
    • 4y + 3 − 3 ≤ −5 − 3
  • Simplify the left hand inequality (−2y + 7 − 7 < 1 −7)
    • −2y < __blank
    • −2y < −6
  • Simplify the right side inequality (4y + 3 − 3 ≤ −5 − 3)
    • 4y ≤ __blank
    • 4y ≤ −8
  • In the left hand inequality (−2y + 7 < 1), divide both sides by −2 to isolate the y variable. In the right side inequality (4y + 3 ≤ −5), divide both sides by 4 to isolate the y variable.
    fraction negative 2 over 2 less than fraction negative 6 over 2 or fraction 4y over 4 less than or equal to fraction negative 8 over 4
  • Simplify the lefthand inequality. Flip the sign when dividing by a negative.
    • y > __blank
    • y > 3
  • Simplify the right side inequality
    • y ≤ __blank
    • y ≤ −2
  • Now let's graph our compound inequality y > 3 or y ≤ −2. Construct the graph for these solutions. Remember to use an open or closed circle where appropriate.
    blank number line from negative 8 to 8
  • Answer:
    number line from negative 8 to 8 with a closed circle on negative 2 and an arrow going left and an open circle on 3 with an arrow going right

You've completed these review activities!