(1) Create a word problem for the inequality: [tex]$x \leq 3x+4$[/tex]
(2) Use the idea of balancing to solve the following simple linear inequalities.
i. [tex]$2x-13\ \textgreater \ 29$[/tex]
ii. [tex]$5x-9\ \textgreater \ -5$[/tex]
iii. [tex]$15\ \textless \ 8-2x$[/tex]
iv. [tex]$3x \leq 8+x$[/tex]
v. [tex]$x-55\ \textgreater \ 14$[/tex]
vi. [tex]$13-x^2\ \textless \ 12$[/tex]
vii. [tex]$x-3 \geq 2$[/tex]
viii. [tex]$2x^2-5 \leqslant 35-x^2$[/tex]

Create a word problem for the inequality x ≤ 3 x + 4 .
Solve the linear inequalities by isolating x using algebraic manipulations.
The solutions are: i) 21"> x > 21 , ii) \frac{4}{5}"> x > 5 4 ​ , iii) x < − 2 7 ​ , iv) x ≤ 4 , v) 69"> x > 69 , vi) 1"> x > 1 or x < − 1 , vii) 25"> x > 25 , viii) − 3 40 ​ ​ ⩽ x ⩽ 3 40 ​ ​ .
The solutions to the inequalities are presented above. 21, x > \frac{4}{5}, x < -\frac{7}{2}, x \leq 4, x > 69, x > 1 \text{ or } x < -1, x > 25, -\sqrt{\frac{40}{3}} \leqslant x \leqslant \sqrt{\frac{40}{3}}}"> x > 21 , x > 5 4 ​ , x < − 2 7 ​ , x ≤ 4 , x > 69 , x > 1 or x < − 1 , x > 25 , − 3 40 ​ ​ ⩽ x ⩽ 3 40 ​ ​ ​

Explanation

Creating a Word Problem We are given the inequality x ≤ 3 x + 4 and several other inequalities to solve. First, let's create a word problem for the inequality x ≤ 3 x + 4 .

Word Problem Here's a word problem for x ≤ 3 x + 4 :


'You have a certain amount of money, x dollars. You want to buy some items. If you triple your money and add $4, you would have more than or equal to what you have now. What is the minimum amount of money you could have?'

Solving Inequality i Now, let's solve the given inequalities:

i. 29"> 2 x − 13 > 29 Add 13 to both sides: 29 + 13"> 2 x > 29 + 13 , so 42"> 2 x > 42 .
Divide both sides by 2: \frac{42}{2}"> x > 2 42 ​ , so 21"> x > 21 .

Solving Inequality ii ii. -5"> 5 x − 9 > − 5 Add 9 to both sides: -5 + 9"> 5 x > − 5 + 9 , so 4"> 5 x > 4 .
Divide both sides by 5: \frac{4}{5}"> x > 5 4 ​ .

Solving Inequality iii iii. 15 < 8 − 2 x Subtract 8 from both sides: 15 − 8 < − 2 x , so 7 < − 2 x .
Divide both sides by -2 (and flip the inequality sign): x"> − 2 7 ​ > x , so x < − 2 7 ​ .

Solving Inequality iv iv. 3 x ≤ 8 + x Subtract x from both sides: 3 x − x ≤ 8 , so 2 x ≤ 8 .
Divide both sides by 2: x ≤ 2 8 ​ , so x ≤ 4 .

Solving Inequality v v. 14"> x − 55 > 14 Add 55 to both sides: 14 + 55"> x > 14 + 55 , so 69"> x > 69 .

Solving Inequality vi vi. 13 − x 2 < 12 Subtract 13 from both sides: − x 2 < 12 − 13 , so − x 2 < − 1 .
Multiply both sides by -1 (and flip the inequality sign): 1"> x 2 > 1 .
This means 1"> x > 1 or x < − 1 .

Solving Inequality vii vii. 22"> x − 3 > 22 Add 3 to both sides: 22 + 3"> x > 22 + 3 , so 25"> x > 25 .

Solving Inequality viii viii. 2 x 2 − 5 ⩽ 35 − x 2 Add x 2 and 5 to both sides: 2 x 2 + x 2 ⩽ 35 + 5 , so 3 x 2 ⩽ 40 .
Divide both sides by 3: x 2 ⩽ 3 40 ​ .
Taking the square root of both sides: − 3 40 ​ ​ ⩽ x ⩽ 3 40 ​ ​ .
Since 3 40 ​ ​ ≈ 3.65 , we have approximately − 3.65 ⩽ x ⩽ 3.65 .

Final Solutions In summary, the solutions to the inequalities are:


i. 21"> x > 21 ii. \frac{4}{5}"> x > 5 4 ​ iii. x < − 2 7 ​ iv. x ≤ 4 v. 69"> x > 69 vi. 1"> x > 1 or x < − 1 vii. 25"> x > 25 viii. − 3 40 ​ ​ ⩽ x ⩽ 3 40 ​ ​
Examples
Linear inequalities are used in everyday life to make decisions when there are constraints. For example, suppose you want to buy a certain number of apples and oranges, but you only have a limited budget. You can use linear inequalities to determine the possible combinations of apples and oranges you can buy without exceeding your budget. Similarly, businesses use linear inequalities to optimize production costs and maximize profits, considering constraints such as available resources and market demand. Understanding and solving linear inequalities helps in making informed decisions in various real-world scenarios.

Answered by GinnyAnswer | 2025-07-05