Solving Multi-Step Inequalities

Solving Multi-Step Inequalities exercise

• How to set up an inequality.

  Hints

  Think about a way to represent the number of workout sessions.

  For example:

  • If Rocky does $4$ workout sessions during the $10$ days, he can loses $4\times 5$ pounds.

  Solution

  When Rocky won last year's beauty competition,he weighed 50 pounds. Just to qualify for this year's competition, he must lose at least $20$ pounds. He has $10$ days to lose the weight.

  • He can lose $5$ pounds with each workout session. We can express this as $-5x$.
  • Since Rocky eats $2$ lbs. of meatballs per day, we can write this as a constant: $2\times 10$.
  • Because Rocky has to lose at least $20$ pounds, we use the $\le$ sign. So our answer, $x$, will be the minimum number of workout sessions needed to lose $20$ pounds.

  $-5x+(2\times 10)\le -20$

• Determine the number of workout sessions Rocky needs to reach his goal of losing $20$ pounds.

  Hints

  You can solve the inequality above just like you would solve an equation. First, combine like terms then use opposite operations to isolate the variable $x$.

  Remember:

  • If you multiply or divide an inequality by a negative number, you have to flip the inequality sign.

  • ($\le~\longleftrightarrow~\ge$)

  For example: The statement $2 \le 4$ is correct. If we multiply this statement by $-2$, we get $-4 \ge -8$.

  Solution

  Let's look at the given facts:

  • Rocky loses $5$ pounds per workout session
  • He also eats $2$ pounds of meatballs per day for $10$ days
  • Rocky wants to lose at least $20$ pounds.

  To express this mathematically, we use an inequality symbol ($\le$ or $\ge$). For this problem, we should use the less than or equal to sign ($\le$).

  $~$

  We can write the inequality: ($-5 = $ weight Rocky loses per workout session)($x= $ mumber of workout sessions) + (2 pounds of meatballs)(10 days) $\le$ ($-20 = $ Rocky's target weight loss)

  $~$

  $\begin{array}{rcll} -5x + (2 \times 10) &\le& -20& \\ -5x + 20 &\le& -20& \\ \color{#669900}{-20} && \color{#669900}{-20} &~~~~~|~ \text{Opposite Operations} \\ -5x &\le& -40 \\ \color{#669900}{\div{-5}} && \color{#669900}{\div{-5}} &~~~~~|~ \text{Opposite Operations} \\ x &{\ge}& 8 &~~~~~|~ \text{Flip the sign} \le~ \rightarrow ~\ge \text{because we divide by a negative number} \end{array}$

  $~$

  This means that Rocky should participate in at least eight workout sessions. Of course, he can also participate in more than eight sessions.

  $~$

  Let's check our work:

  $8$ sessions:

  $\begin{array}{rcll} -5(8) + 20 &\le& -20 &\\ -20 &\le& -20 &~~~~~\surd \end{array}$

  $10$ sessions:

  $\begin{array}{rcll} -5(10) + 20 &\le& -20 & \\ -50 + 20 &\le& -20 & \\ -30 &\le& -20 &~~~~~\surd \end{array}$

  Go Rocky!

• Decide how many workout sessions Bruno needs to complete.

  Hints

  First, determine the inequality.

  The inequality is: $-4x+(3\times 10)\le -10$.

  Solve this inequality as follows:

  1. Simplify
  2. Combine Like Terms
  3. Solve for $x$

  Remember, because you divide by a negative number across the inequality, the inequality sign has to change directon.

  Solution

  What do we know?

  • Bruno loses $4$ pounds per workout session.
  • Bruno eats $3$ pounds of meatballs per day for $10$ days.
  • He wants to lose at least $10$ pounds.

  We can write the inequality:

  $-4x+(3\times 10)\le -10$

  $~$

  $\begin{array}{rcll} -4x + (3 \times 10) &\le& -10& \\ -4x + 30 &\le& -10& \\ \color{#669900}{-30} && \color{#669900}{-30} &~~~~~|~ \text{Opposite Operations} \\ -4x &\le& -40 \\ \color{#669900}{\div{-4}} && \color{#669900}{\div{-4}} &~~~~~|~ \text{Opposite Operations} \\ x &{\ge}& 10 &~~~~~|~ \text{Flip the sign} \le~ \rightarrow ~\ge \text{because we divide by a negative number} \end{array}$

  To lose the amount of weight he wants, Bruno needs to complete at least $10$ workout sessions. Of course, if he enjoys it, he can do more than $10$ sessions.

  Perhaps Rocky and Bruno should be workout partners!

• Examine the effect of different diets on the Rocky's weight loss.

  Hints

  For the inequality, what is the unknown quantity?

  $-5x+(?\times 10)\le -30$

  The ? represent the amount of weight gained after eating each type of food.

  Solve each inequality by

  1. Simplifying
  2. Combining like terms
  3. Solving for $x$ by using opposite operations

  You can also check your answers by plugging the solution back into each inequality.

  Solution

  For this problem set, to solve each inequality, the only change is the weight gain as a result of eating the different food options.

  Salad

  $\begin{array}{rcll} -5x + (0.5 \times 10) &\le& -30& \\ -5x + 5 &\le& -30& \\ \color{#669900}{-5} && \color{#669900}{-5} &~~~~~|~ \text{Opposite Operations} \\ -5x &\le& -35 \\ \color{#669900}{\div{-5}} && \color{#669900}{\div{-5}} &~~~~~|~ \text{Opposite Operations} \\ x &{\ge}& 7 &~~~~~|~ \text{Flip the sign} \le~ \rightarrow ~\ge \text{because we divide by a negative number} \end{array}$

  $~$

  Sausages

  $\begin{array}{rcll} -5x + (1 \times 10) &\le& -30& \\ -5x + 10 &\le& -30& \\ \color{#669900}{-10} && \color{#669900}{-10} &~~~~~|~ \text{Opposite Operations} \\ -5x &\le& -40 \\ \color{#669900}{\div{-5}} && \color{#669900}{\div{-5}} &~~~~~|~ \text{Opposite Operations} \\ x &{\ge}& 8 &~~~~~|~ \text{Flip the sign} \le~ \rightarrow ~\ge \text{because we divide by a negative number} \end{array}$

  $~$

  Steak

  $\begin{array}{rcll} -5x + (1.5 \times 10) &\le& -30& \\ -5x + 15 &\le& -30& \\ \color{#669900}{-15} && \color{#669900}{-15} &~~~~~|~ \text{Opposite Operations} \\ -5x &\le& -45 \\ \color{#669900}{\div{-5}} && \color{#669900}{\div{-5}} &~~~~~|~ \text{Opposite Operations} \\ x &{\ge}& 9 &~~~~~|~ \text{Flip the sign} \le~ \rightarrow ~\ge \text{because we divide by a negative number} \end{array}$

  $~$

  Meatballs

  $\begin{array}{rcll} -5x + (2 \times 10) &\le& -30& \\ -5x + 20 &\le& -30& \\ \color{#669900}{-20} && \color{#669900}{-20} &~~~~~|~ \text{Opposite Operations} \\ -5x &\le& -50 \\ \color{#669900}{\div{-5}} && \color{#669900}{\div{-5}} &~~~~~|~ \text{Opposite Operations} \\ x &{\ge}& 10 &~~~~~|~ \text{Flip the sign} \le~ \rightarrow ~\ge \text{because we divide by a negative number} \end{array}$

• Describe the steps required to solve the inequality $-5x+(2\times5)\le -20$.

  Hints

  Use opposite operations:

  • Multiplication ($\times~\longleftrightarrow~\div$)
  • Division ($\div~\longleftrightarrow~\times$)
  • Addition ($+~\longleftrightarrow~-$)
  • Subtraction ($-~\longleftrightarrow~+$)

  Don't forget: when multiplying by or dividing by a negative number across an inequality, you have to remember to flip the inequality sign.

  Solution

  $\begin{array}{rcll} -5x + (2 \times 10) &\le& -20& \\ -5x + 20 &\le& -20&~~~~~|~ \text{Simplify by Multiplying} \\ \color{#669900}{-20} && \color{#669900}{-20} &~~~~~|~ \text{Combine Like Terms using Opposite Operations} \\ -5x &\le& -40 \\ \color{#669900}{\div{-5}} && \color{#669900}{\div{-5}} &~~~~~|~ \text{Opposite Operations} \\ x &{\ge}& 8 &~~~~~|~ \text{Flip sign} \le~ \rightarrow ~\ge \text{because we divided by a negative number} \end{array}$

• Solve the given inequalities.

  Hints

  You can solve each equation by:

  • Simplifying
  • Combining Like Terms
  • Solving for $x$

  Use the opposite operations:

  • Multiplication ($\times~\longleftrightarrow~\div$)
  • Division ($\div~\longleftrightarrow~\times$)
  • Addition ($+~\longleftrightarrow~-$)
  • Subtraction ($-~\longleftrightarrow~+$)

  Solution

  How do we solve inequalities?

  1. First, we simplify, if it's necessary.
  2. Next, we Combine Like Terms using the Opposite Operations of addition or subtraction.
  3. Finally, we use Opposite Operations like multiplication or division to solve for $x$.

  Important: If you multiply or divide by a negative number, you have to change the direction of the inequality sign.

  $\begin{array}{rcll} 4x + 4 &\ge& 12& \\ \color{#669900}{-4} && \color{#669900}{-4} &~~~~~|~ \text{Opposite Operations} \\ 4x &\ge& 8 \\ \color{#669900}{\div{4}} && \color{#669900}{\div{4}} &~~~~~|~ \text{Opposite Operations} \\ x &\ge& 2 & \end{array}$

  $~$

  $\begin{array}{rcll} -2x + (5 \times 3) &\ge& 5& \\ -2x + 15 &\ge& 5& \\ \color{#669900}{-15} && \color{#669900}{-15} &~~~~~|~ \text{Opposite Operations} \\ -2x &\ge& -10 \\ \color{#669900}{\div{-2}} && \color{#669900}{\div{-2}} &~~~~~|~ \text{Opposite Operations} \\ x &\color{#669900}{\le}& 5 &~~~~~|~ \text{Flip the sign} \ge~ \rightarrow ~\le \text{because we divide by a negative number} \end{array}$

  $~$

  $\begin{array}{rcll} 3x - (2 \times 4) &\le& 4& \\ 3x - 8 &\le& 4& \\ \color{#669900}{+8} && \color{#669900}{+8} &~~~~~|~ \text{Opposite Operations} \\ 3x &\le& 12 \\ \color{#669900}{\div{3}} && \color{#669900}{\div{3}} &~~~~~|~ \text{Opposite Operations} \\ x &\le& 4 \end{array}$

  $~$

  $\begin{array}{rcll} -6x - (3 \times 12) &\le& -60& \\ -6x - 36 &\le& -60& \\ \color{#669900}{+36} && \color{#669900}{+36} &~~~~~|~ \text{Opposite Operations} \\ -6x &\le& -24 \\ \color{#669900}{\div{-6}} && \color{#669900}{\div{-6}} &~~~~~|~ \text{Opposite Operations} \\ x &\color{#669900}{\ge}& 4 &~~~~~|~ \text{Flip the sign} \le~ \rightarrow ~\ge \text{because we divide by a negative number} \end{array}$