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SOLVING LINEAR INEQUALITIES, ALL MIXED UP
Remember:
If you multiply or divide both sides of an inequality
by a negative number,
then you must change the direction of
the inequality symbol.
EXAMPLES:
Solve:
[beautiful math coming... please be patient]
$-6 - 3x \ge 4$
Solution:
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$-6 - 3x \ge 4$
(original sentence)
[beautiful math coming... please be patient]
$-3x \ge 10$
(add $\,6\,$ to both sides)
[beautiful math coming... please be patient]
$x \le -\frac{10}{3}$
(divide both sides by $\,-3\,$;
change the direction of the inequality symbol)
Solve:
[beautiful math coming... please be patient]
$3 - 2x \le 5x + 1$
Solution:
[beautiful math coming... please be patient]
$3 - 2x \le 5x + 1$
(original sentence)
[beautiful math coming... please be patient]
$3 - 7x \le 1$
(subtract $\,5x\,$ from both sides)
[beautiful math coming... please be patient]
$-7x \le -2$
(subtract $\,3\,$ from both sides)
[beautiful math coming... please be patient]
$x \ge \frac{2}{7}$
(divide both sides by $\,-7\,$; change the direction of the inequality symbol)
Solve:
[beautiful math coming... please be patient]
$\displaystyle -\frac{2}{3}x + 6\le 1$
Solution:
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$\displaystyle -\frac{2}{3}x + 6\le 1$
(original sentence)
[beautiful math coming... please be patient]
$-2x + 18\le 3$
(clear fractions; multiply both sides by $\,3\,$)
[beautiful math coming... please be patient]
$-2x \le -15$
(subtract $\,18\,$ from both sides)
[beautiful math coming... please be patient]
$\displaystyle x \ge \frac{15}{2}$
(divide both sides by $\,-2\,$; change the direction of the inequality symbol)
For more advanced students, a graph is displayed.
For example, the inequality $3 - 2x \le 5x + 1$
is optionally accompanied by the
graph of $\,y = 3 - 2x\,$ (the left side of the inequality, dashed green)
and the graph of
$\,y = 5x + 1\,$ (the right side of the inequality, solid purple).
In this example, you are finding the values of $\,x\,$ where the green
graph lies on or below the purple graph.
Click the “show/hide graph” button if you prefer not to see the graph.