Solving One-Step Inequalities by Multiplying or Dividing

Solving One-Step Inequalities by Multiplying or Dividing exercise

• Explain the rules of the cavemen's game.

  Hints

  Think about all the signs that can be flipped.

  Does it make sense to flip the equal sign?

  For the inequality $-2x \le 8$, look what happens when we isolate the variable $x$.

  $\begin{array}{rcrcl} -2x &\le& 8\\ \color{#669900}{\div(-2)}&&\color{#669900}{\div(-2)}&&| (\times~\leftrightarrow~\div) \text{ Opposite Operations}\\ x&\ge&-4&&|\text{ Look what happened to the Inequality Sign!} \end{array}$

  Do you see the problem? Which operation did we perform?

  $\begin{array}{rcr} -2 &~~<& 5\\ \color{#669900}{\times (-1)} &~~~& \color{#669900}{\times (-1)}\\ 2 &~~<& -5 \end{array}$

  Solution

  When solving inequalities you must pay special attention to the inequality sign.

  But don't worry and just remember: When you multiply or divide an inequality by a negative number, you must flip the inequality sign.

  To see how it works, let's solve this inequality:

  $\begin{array}{rcr} -2 &~~<& 5\\ \color{#669900}{\times (-1)} &~~~& \color{#669900}{\times (-1)}\\ 2 &~~>& -5 \end{array}$

  As you can see, the multiplication of a negative number calls for a flip of the inequality sign. Otherwise the result would be $2<-5$, which is not correct.

• Calculate the number of times at least the first cavemen hit the bullseye.

  Hints

  When calculating with inequalities, remember that the solution set also includes numbers greater than or less than the final solution.

  The inequality $-2 < x$ has an infinite number of solutions. There are countless integers larger than $-2$.

  Solution

  Let's take a deeper look at the score of the first cavemen - who has $12$ points.

  What does this score mean to us? Since we don't know the play-by-play of the game, we will have to make some assumptions:

  • If we assume the caveman never missed the target, we can use the following equation: $3x =12$. The variable $x$ stands for the number of times the caveman hit the bullseye. We can solve this equation by dividing both sides by $3$. The solution is $x=4$. This means the caveman threw the bone four times and never once missed the target.
  • If we assume the caveman had some misses as well, we can use the inequality $3x \ge 12$. If we by divide by $3$, we are left with $x \ge 4$. In this case the caveman hit the bullseye four or more times.

• Analyze the scores of the cavemen's friends.

  Hints

  Pay attention, and read the problem carefully. To figure out the number of attempts, depending if the caveman had any hits at all, you may use an equation or an inequality.

  To solve this equation, see how we flip the inequality sign when we multiply by a negative number?

  \begin{array}{rcr} -\frac12 x &~~\le& -5\\ \color{#669900}{\times (-2)} &~~~& \color{#669900}{\times (-2)}\\ x &~~\ge& 10 \end{array}

  Solution

  These cavemen don't really seem to be that good at throwing. Each has a negative score! What conclusions can we draw from the results of their competition?

  • With a score of $-4$ points, we can write the equation $- \frac12 x \le -4$. Dividing by $-\frac12$ means multiplying by $(-2)$. We must remember to flip the inequality sign. Now we get $x \ge 8$. So this caveman had at least $8$ throws.
  • Another caveman had a score of $-7$ points. Since he had no successful throws, our equation is $-\frac12 x = -7$. Multiplying both sides by $(-2)$ gives us $x=14$. This caveman had exactly $14$ throws since he never hit the bullseye.
  • Another not-very-talented caveman has a score of $-2$. Calculating as before, we transform our inequality, $-\frac12 x \le -2$, to $x \ge 4$. This caveman had at least four throws.
  • The last caveman had the best score of $-1$. This means he had at least two throws.

• Decide which sign has to be flipped to solve the inequality.

  Hints

  Our goal is to isolate $x$. This way, we are able to define the solution set of $x$.

  Solution

  Remember, when multiplying or dividing an inequality by a negative number, flip the sign.

  To solve these inequalities, flip the sign:

  • $-3x<5$
  • $-4x<-7$
  • $-x \le 1$
  • $-x \ge 0$

  To solve these inequalities, maintain the sign:

  • $4x>-3$
  • $2x \ge 0$

• Define the meaning of each inequality sign.

  Hints

  Which inequality sign makes a true statement?

  • $-2<5$
  • $-2 > 5$
  • $-2 \le 5$
  • $-2 \ge 5$

  When we introduce variables, we get even bigger solution sets.

  Which numbers fulfill the inequality $x \le 5$?

  Solution

  There are four inequality signs:

  • $x<y$ meaning $x$ less than $y$
  • $x>y$ meaning $x$ greater than $y$
  • $x\le y$ meaning $x$ less or equal to $y$
  • $x\ge y$ meaning $x$ greater or equal to $y$

  Solving inequality problems requires a little more attention than solving equations because when you multiply or divide by a negative number, you must flip the inequality sign.

• Calculate the inequality $-2x-2 \ge (-2x+1) \times 4$.

  Hints

  When you multiply or divide an inequality by an negative number, you must flip the inequality sign.

  Solution

  To solve this problem, there is more than one way to combine like terms. For the method shown here, the sign does not flip, but there is an alternative sequence of steps that results in flipping the sign:

  Beginning with row $3$ we could have transformed $-2x \ge -8x+6$ to $-6 \ge -6x$. Then we would have divided by $-6$ getting $1 \le x$, which is the same answer.