Answer :
We start by solving each inequality step by step.
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1. For the inequality
[tex]$$4x - 23 < 9,$$[/tex]
first add 23 to both sides:
[tex]$$4x < 9 + 23 = 32.$$[/tex]
Then divide both sides by 4:
[tex]$$x < \frac{32}{4} = 8.$$[/tex]
Thus, the solution for this inequality is
[tex]$$x < 8.$$[/tex]
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2. Next, consider the inequality
[tex]$$5x + 42 \geq 57.$$[/tex]
Subtract 42 from both sides:
[tex]$$5x \geq 57 - 42 = 15.$$[/tex]
Now, divide both sides by 5:
[tex]$$x \geq \frac{15}{5} = 3.$$[/tex]
So, the solution is
[tex]$$x \geq 3.$$[/tex]
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3. Finally, solve
[tex]$$\frac{x}{3} + 6 > 42.$$[/tex]
Subtract 6 from both sides:
[tex]$$\frac{x}{3} > 42 - 6 = 36.$$[/tex]
Multiply both sides by 3:
[tex]$$x > 36 \times 3 = 108.$$[/tex]
Thus, the solution is
[tex]$$x > 108.$$[/tex]
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To summarize, the solutions are:
- For [tex]$4x - 23 < 9$[/tex]: [tex]$$x < 8.$$[/tex]
- For [tex]$5x + 42 \geq 57$[/tex]: [tex]$$x \geq 3.$$[/tex]
- For [tex]$\frac{x}{3}+6>42$[/tex]: [tex]$$x > 108.$$[/tex]
You can now match these solutions to the corresponding number lines.
________________________________
1. For the inequality
[tex]$$4x - 23 < 9,$$[/tex]
first add 23 to both sides:
[tex]$$4x < 9 + 23 = 32.$$[/tex]
Then divide both sides by 4:
[tex]$$x < \frac{32}{4} = 8.$$[/tex]
Thus, the solution for this inequality is
[tex]$$x < 8.$$[/tex]
________________________________
2. Next, consider the inequality
[tex]$$5x + 42 \geq 57.$$[/tex]
Subtract 42 from both sides:
[tex]$$5x \geq 57 - 42 = 15.$$[/tex]
Now, divide both sides by 5:
[tex]$$x \geq \frac{15}{5} = 3.$$[/tex]
So, the solution is
[tex]$$x \geq 3.$$[/tex]
________________________________
3. Finally, solve
[tex]$$\frac{x}{3} + 6 > 42.$$[/tex]
Subtract 6 from both sides:
[tex]$$\frac{x}{3} > 42 - 6 = 36.$$[/tex]
Multiply both sides by 3:
[tex]$$x > 36 \times 3 = 108.$$[/tex]
Thus, the solution is
[tex]$$x > 108.$$[/tex]
________________________________
To summarize, the solutions are:
- For [tex]$4x - 23 < 9$[/tex]: [tex]$$x < 8.$$[/tex]
- For [tex]$5x + 42 \geq 57$[/tex]: [tex]$$x \geq 3.$$[/tex]
- For [tex]$\frac{x}{3}+6>42$[/tex]: [tex]$$x > 108.$$[/tex]
You can now match these solutions to the corresponding number lines.
