Given:
The inequalities are:
[tex]2x-3<7[/tex]
[tex]2x+1>-3x-4[/tex]
To find:
The range of values of [tex]x[/tex] for the given inequalities.
Solution:
We have,
[tex]2x-3<7[/tex]
Adding 3 on both sides, we get
[tex]2x-3+3<7+3[/tex]
[tex]2x<10[/tex]
Divide both sides by 2.
[tex]\dfrac{2x}{2}<\dfrac{10}{2}[/tex]
[tex]x<5[/tex] ...(i)
The second inequality is:
[tex]2x+1>-3x-4[/tex]
Subtracting 1 from both sides, we get
[tex]2x+1-1>-3x-4-1[/tex]
[tex]2x>-3x-5[/tex]
Adding [tex]3x[/tex] on both sides, we get
[tex]2x+3x>-3x-5+3x[/tex]
[tex]5x>-5[/tex]
Divide both sides by 5.
[tex]\dfrac{5x}{5}>\dfrac{-5}{5}[/tex]
[tex]x>-1[/tex] ...(ii)
Using (i) and (ii), we get
[tex]-1<x<5[/tex]
Therefore, the required range is [tex]-1<x<5[/tex].
