Solve the given inequality algebraically: 3 x + 1 ≥ 2 x + 5

Given, 3 x + 1 ≥ 2 x + 5 . . . . . i We need to find the solution of the given inequality. Adding - 1 to both sides of the inequality i , we have 3 x + 1 - 1 ≥ 2 x + 5 - 1 ⇒ 3 x ≥ 2 x + 4 Adding ' - 2 x ' to both sides of the above inequality, we have 3 x - 2 x ≥ 2 x + 4 - 2 x ⇒ x ≥ 4 Hence, all real numbers x which are greater than or equal to 4 are the solutions of the given inequality i.e., x ∈ [ 4 , ∞ ) .

Solve the given inequality algebraically: 10 ≤ 2 x ≤ x + 9

Given, 10 ≤ 2 x ≤ x + 9 Thus, the system of inequalities we have are as follows: 2 x ≥ 10 . . . . i 2 x ≤ x + 9 . . . . i i Now dividing each side of the inequality i by 2 , we have x ≥ 5 . . . i i i Now, consider 2 x ≤ x + 9 Adding ' - x ' to both sides of the above inequality, we have 2 x - x ≤ x + 9 - x ⇒ x ≤ 9 . . . . . i v We know that the solution of the system of inequalities is the intersection region of the solutions of the two inequalities. Hence, from i i i , and i v The solution set for the given inequality is 5 ≤ x ≤ 9

Solve the given inequality graphically 4 - 2 x ≤ 2

Given inequality is 4 - 2 x ≤ 2 We need to find the solution of this inequality graphically i.e., we need to find all those values of x for which the given inequality is a true statement. For that let f 1 x = 4 - 2 x , and f 2 x = 2 Let us draw the lines f 1 x = 4 - 2 x , and f 2 x = 2 , and find where they intersect. Consider the following table x 0 2 f 1 x = 4 - 2 x 4 - 2 0 = 4 0 Thus, f 1 x = 4 - 2 x is a straight line obtained by joining the points 0 , 4 , and 2 , 0 . And, it is represented by a solid line in the x y -plane. Consider f 2 x = 2 We know that this line is a horizontal straight line passing through the point 0 , 2 . And we need to represent this line using a solid line to show that the points on the line f 2 x = 2 are also the solution for the given inequality as 4 - 2 x ≤ 2 means 4 - 2 x also take the value 2 . Hence, the graph for both the considered functions is given as follows: From the above graph, we find that both the graphs intersect at 1 , 2 i.e, at 1 , 2 , we have 4 - 2 x = 2 . But we need to find the values of x for which 4 - 2 x ≤ 2 . From the graph, the line for f 1 x is below f 2 x when x ≥ 1 Hence, the solution for the given inequality is x ≥ 1 .

Solve the given inequality graphically x 3 + 5 < 11

Given inequality is x 3 + 5 < 11 We need to find the solution of this inequality graphically i.e., we need to find all those values of x for which the given inequality is a true statement. For that let f 1 x = x 3 + 5 , and f 2 x = 11 Let us draw the lines f 1 x = x 3 + 5 , and f 2 x = 11 , and find where they intersect. Consider the following table x 0 - 15 f 1 x = x 3 + 5 5 0 Thus, f 1 x = x 3 + 5 is a straight line joining the points 0 , 5 , and - 15 , 0 . And, we represent it by a solid line in the x y -plane. Consider f 2 x = 11 We know that this line is a horizontal straight line passing through the point 0 , 11 . And we need to represent this line using a dashed line to show that the points on the line f 2 x = 11 are not the possible solutions as x 3 + 5 < 11 means x 3 + 5 cannot take the value 11 . Hence, the graph for both the considered functions is given as follows: From the above graph, we find that both the graphs intersect at 18 , 11 i.e, at 18 , 11 , we have x 3 + 5 = 11 . But we need to find the values of x for which x 3 + 5 < 11 . From the graph, the line for f 1 x is below f 2 x when x < 18 Hence, the solution for the given inequality is x < 18 .