Solve the inequality.
$$3x - 5
- 2 < -2, \, x \, \in \, \mathbb{N}$$
Mark the solution on the number line given below.
John and Gemma played a new computer game ... show full transcript
Worked Solution & Example Answer:Solve the inequality - Junior Cycle Mathematics - Question 12 - 2012
Step 1
Solve the inequality: $3x - 5 < -2, \, x \, \in \, \mathbb{N}$
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Answer
To solve the inequality, we first isolate x:
Add 5 to both sides:
3x<3
Divide both sides by 3:
x<1
Since x must be a natural number, the solution is x=1.
Step 2
Make an equation to represent Gemma's score.
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Answer
Gemma's total score can be represented by the equation
5x−5y=20
where x represents the number of benga scored and y represents the penalties incurred.
Step 3
Use simultaneous equations to find the number of points for a benga and the number of points for a penalty.
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Answer
We have the equations:
2x−3y=7 (John's score)
5x−5y=20 (Gemma's score)
To solve them simultaneously:
From the second equation, simplify by dividing everything by 5:
x−y=4→y=x−4
Substitute into the first equation:
2x−3(x−4)=72x−3x+12=7−x+12=7x=5
Then substitute back to find y:
y=5−4=1.
So the number of points for a benga is 5 and for a penalty is 1.
Step 4
Verify your solutions in both equations.
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Answer
For John's equation:
2(5)−3(1)=710−3=77=7 (True)
For Gemma's equation:
5(5)−5(1)=2025−5=2020=20 (True)
Both equations hold, confirming the solutions are correct.
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