A)
Starting with the inequality:
[tex]-4x+3<11[/tex]
Substract 3 from both sides of the inequality:
[tex]\begin{gathered} -4x<11-3 \\ \Rightarrow-4x<8 \end{gathered}[/tex]
Divide both sides of the inequality by -4. Since -4 is a negative number, swap the < sign:
[tex]\begin{gathered} -\frac{4x}{-4}>\frac{8}{-4} \\ \Rightarrow x>-2 \end{gathered}[/tex]
Plot the relation -2 on a number line:
B)
Use similar arguments, be careful of dividing by negative numbers, as that will change the orientation of the symbol "<" or ">".
This time, the final expression is:
[tex]x<-2[/tex]
Plot in the number line:
C)
Divide both sides by -4 to get:
[tex]x=-3[/tex]
Plot by drawing a single point at x=-3:
Recall the properties of inequalities:
Let a, b and c be real numbers.
If:
[tex]aThen:[tex]\begin{gathered} b>a \\ a+c-b \end{gathered}[/tex]
If c is positive, then:
[tex]\begin{gathered} a\cdot cIf
c is negative, then:[tex]\begin{gathered} a\cdot c>b\cdot c \\ \frac{a}{c}>\frac{b}{c} \end{gathered}[/tex]
In other words: the orientation of the inequality symbol does not change when adding or substracting quantities, and remains the same when multiplying or dividing by a positive number.
The orientation of the inequality symbol changes when the inequality is multiplied or divided by a negative number. This includes multiplying by -1. When writing the inequality by taking the right side to the left and vice versa, the inequality symbol should also swap.
