Find the values of $x$ satisfying:
(a) $|x-4|>|x+1|$
(b) $|x-3|>|x+2|$
(c) $|3 x-2| \geq 3|x-1|$
(d) $|x-5|-|x-3| \geq 0$

Show graphically the regions represented by the inequalities:
(a) $4 x+y-3<0$
(b) $2 x-3 y+1 \leq 0$
(c) $x+2 y+4<0$
(d) $5 y+x-7 \leq 0$

Solve |x+1|"> ∣ x − 4∣ > ∣ x + 1∣ by squaring both sides, resulting in x < 1.5 .
Solve |x+2|"> ∣ x − 3∣ > ∣ x + 2∣ by squaring both sides, resulting in x < 0.5 .
Solve ∣3 x − 2∣ ≥ 3∣ x − 1∣ by squaring both sides, resulting in x ≥ 6 5 ​ .
Solve ∣ x − 5∣ − ∣ x − 3∣ ≥ 0 by squaring both sides, resulting in x ≤ 4 .
Graphically represent each inequality by plotting the corresponding line and shading the appropriate region. The solutions are: (a) x < 1.5 , (b) x < 0.5 , (c) x ≥ 6 5 ​ , (d) x ≤ 4 .

Explanation

Problem Analysis We are given four inequalities involving absolute values and one set of inequalities to represent graphically. Our goal is to solve each inequality for x and represent the regions defined by the inequalities graphically.

Solving Inequality (a) To solve |x+1|"> ∣ x − 4∣ > ∣ x + 1∣ , we square both sides to eliminate the absolute values: (x+1)^2"> ( x − 4 ) 2 > ( x + 1 ) 2
x^2+2x+1"> x 2 − 8 x + 16 > x 2 + 2 x + 1
-15"> − 10 x > − 15
x < 10 15 ​
x < 2 3 ​
x < 1.5

Solving Inequality (b) To solve |x+2|"> ∣ x − 3∣ > ∣ x + 2∣ , we square both sides: (x+2)^2"> ( x − 3 ) 2 > ( x + 2 ) 2
x^2+4x+4"> x 2 − 6 x + 9 > x 2 + 4 x + 4
-5"> − 10 x > − 5
x < 10 5 ​
x < 2 1 ​
x < 0.5

Solving Inequality (c) To solve ∣3 x − 2∣ ≥ 3∣ x − 1∣ , we square both sides: ( 3 x − 2 ) 2 ≥ 9 ( x − 1 ) 2
9 x 2 − 12 x + 4 ≥ 9 ( x 2 − 2 x + 1 )
9 x 2 − 12 x + 4 ≥ 9 x 2 − 18 x + 9
6 x ≥ 5
x ≥ 6 5 ​

Solving Inequality (d) To solve ∣ x − 5∣ − ∣ x − 3∣ ≥ 0 , we can rewrite it as ∣ x − 5∣ ≥ ∣ x − 3∣ . Squaring both sides: ( x − 5 ) 2 ≥ ( x − 3 ) 2
x 2 − 10 x + 25 ≥ x 2 − 6 x + 9
− 4 x ≥ − 16
x ≤ 4

Graphical Representations For the graphical representations: (a) 4 x + y − 3 < 0 can be rewritten as y < − 4 x + 3 . This represents the region below the line y = − 4 x + 3 .
(b) 2 x − 3 y + 1 ≤ 0 can be rewritten as y ≥ 3 2 ​ x + 3 1 ​ . This represents the region above the line y = 3 2 ​ x + 3 1 ​ .
(c) x + 2 y + 4 < 0 can be rewritten as y < − 2 1 ​ x − 2 . This represents the region below the line y = − 2 1 ​ x − 2 .
(d) 5 y + x − 7 ≤ 0 can be rewritten as y ≤ − 5 1 ​ x + 5 7 ​ . This represents the region below the line y = − 5 1 ​ x + 5 7 ​ .

Final Answer The solutions to the inequalities are: (a) x < 1.5 (b) x < 0.5 (c) x ≥ 6 5 ​ (d) x ≤ 4


The graphical representations involve shading the appropriate regions relative to the lines defined by the corresponding equations.
Examples
Understanding and solving inequalities is crucial in various real-world applications. For instance, in economics, companies use inequalities to determine the range of prices that maximize profit. In engineering, inequalities are used to ensure that structures can withstand certain loads or stresses. In computer science, inequalities are used in algorithm design to optimize performance. Moreover, graphical representation of inequalities helps in visualizing constraints and feasible regions in optimization problems, making it easier to identify optimal solutions.

Answered by GinnyAnswer | 2025-07-03

The solutions for the inequalities presented are as follows: (a) x < 1.5, (b) x < 0.5, (c) x ≥ 5/6, and (d) x ≥ 4.
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Answered by Anonymous | 2025-07-04