Find the integer values of x that satisfy both of these inequalities.
2 answers:
Answer:
x ∈ [-2/3, 3)
Step-by-step explanation:
5 − 3x ⩽ 7 and 4x + 1 < 13, this is given
-3x ≤ 7-5 and 4x < 13 -1, subtract 5 from first inequality and subtract -1 from second
-3x ≤ 2 and 4x < 12, combine like terms
x ≥ -2/3 and x < 12/4, divide by -3 and change the first inequality, and divide the second inequality by 4
x ≥ -2/3 and x < 3
[-2/3, ∞) and (-∞, 3)
x ∈ [-2/3, 3)
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Remark
There are two ways to do this. You can graph the equations (which I have included at the end of the question using Desmos) or you can solve it algebraically which in this case is going to be slightly painful.
Algebra
Multiply the first equation by 4 and the second by 3
<em><u>First Equation</u></em>
4 [3x + 4y = 16]
12x + 16y = 64 (3)
<em><u>Second Equation</u></em>
3 [-4x - 3y = - 19]
- 12x - 9y = - 57 (4)
Add (3) and (4)
12x + 16y = 64
<u>- 12x - 9y = -57 </u> Add
7y = 7 Divide by 7
y = 1
Solve for x
3x + 4y = 16 Let y = 1
3x + 4(1) = 16
3x + 4 = 16 Subtract 4
3x = 12 Divide by 3
x = 12/3
x = 4
Answer
(4,1) just as the graph suggests.
