Study Guides | inequalities | Solving Inequalities with Variables on Both Sides

Study Guide: Solving Inequalities with Variables on Both Sides

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There are two situation when we need to reverse the direction the inequality symbol:

  1. When reversing the left and right sides of the inequality
  2. When multiplying or dividing both sides by a negative number

1) When reversing the inequality

\begin{align} 12 &\lt x + 5 &&\text{Given.} \tag 1 \\[2ex] 12 {\color{red}- 5} &\lt x + 5 {\color{red}- 5} &&\text{Isolate the variable.} \tag 2 \\[2ex] 7 &\lt x &&\text{Simplified.} \tag 3 \\[2ex] x &\gt 7 &&\text{Solution: Sides reversed.} \tag 4 \end{align}

2) When multiplying or dividing by a negative value

\begin{align} −2y &\lt −8 &&\text{Given.} \tag 1 \\[2ex] \dfrac{-2y}{\color{red}{-2}};&{\color{red}\gt};\dfrac{−8}{\color{red}{-2}} &&\text{Divide by }{\color{red}{-2}}\text{ and reverse the sign.} \tag 2 \\[2ex] y &\gt 4 &&\text{Solution.} \tag 3 \end{align}

Careful!

Don’t multiply or divide by a variable unless you know if it will always be either positive or always negative. If you know it will always be negative, you must also reverse the equality symbol.

Solve each inequality and graph the solution

1) $ 2x \gt 4x - 6 $

<                                                                                     >
$x \lt 3 $

2) $ 7y +1 \le y - 5 $

<                                                                                     >
$ y \le -1 $

3) $ 27x + 33 > 58x - 29 $

<                                                                                     >
$ x \lt 2 $

4) $ -3r \lt 10 - r $

<                                                                                     >
$ r \gt -5 $

5) $ 5c - 4 \gt 8c + 2 $

<                                                                                     >
$ c \lt -2 $

6) $ 4.5x - 3.8 \ge 1.5x - 2.3 $

<                                                                                     >
$ x \ge \frac{1}{2} $

7) The eighth grade class will sell original hand made gift cards sets to raise money for their end of year celebration. Materials will cost $100 plus $4 for each set of gift card. If the class sells the gift cards for $7 per set, how many sets will they need to sell to make a profit?

Logic: ($100) + ($4 x Sets Sold) is less then ($7 x Sets Sold)
Equation: $ 100 + 4s \lt 7s $
Simplified: $ s \gt 33.33 $
Conclusion: They’ll need to sell at least 34 sets to break even.

8) $ 5(4 + x )\le 3(2 + x) $

<                                                                                     >
$ x \le -7 $

9) $ -4(3 - p) \gt 5(p + 1) $

<                                                                                     >
$ p \lt -17 $

10) $ 2(6-x)\lt 4x $

<                                                                                     >
$ x \gt 2 $

11) $ 4x \gt 3(7-x) $

<                                                                                     >
$ x \gt 3 $

12) $ \frac{1}{2}f + \frac{3}{4} \ge \frac{1}{4}f $

<                                                                                     >
$ f \gt -3 $

13) $ -36.72 + 5.65t \lt 0.25t $

<                                                                                     >
$$ t \lt 6.8 $$

Solution for TB Pg 165, 20 - 48 (even)

  1. $$ x \ge -4 $$
  1. $$ x \gt 1 $$
  1. $$ t \gt 1 $$
  1. \begin{align} 12(x+2)&\gt \frac{1}{2}(10)(x+16) \\\ x &\gt 8 \end{align}
  1. $$ n \lt 6 $$
  1. $$ t \lt 3 $$
  1. $$ n \gt 3 $$
  1. $$ \textit{All real numbers} $$
  1. $$ \textit{No solutions} $$
  1. $$ \textit{No solutions} $$
  1. $$ y \gt -4 $$
  1. $$ p \lt 16\frac{1}{3} $$
  1. $$ x \le 4 $$
  1. $$ \textit{All real numbers} $$
  1. $$ n \le \frac{11}{3} $$
Source: https://class.ronliskey.com/study/mathematics-8/8-3-5-solving-inequalities-with-vars-on-both-sides/