# The inequality − 4 &lt; 5 − 3 x ≤ 17 . ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805 ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

(a)

To determine

## To solve: The inequality −4<5−3x≤17.

Expert Solution

The interval of x is [4,3).

### Explanation of Solution

Consider the inequality 4<53x17.

Simplify the above expression as follows,

4<53x1745<3x1759<3x129>3x12(multiplybothsidebyminus1)

3>x4(dividebothsideby3)

Thus, the interval of x is [4,3).

(b)

To determine

### To solve: The inequality x2<2x+8.

Expert Solution

The solution of the interval of x is (2,4).

### Explanation of Solution

Consider the inequality x2<2x+8.

Simplify the above expression as follows,

x2<2x+8x22x8<0x24x+2x8<0x(x4)+2(x4)<0

(x4)(x+2)<0

The inequality (x4)(x+2)<0 will change sign at the critical values x=2andx=4.

The corresponding possible intervals of solution are (,2),(2,4)and(4,).

It is clear that, the interval (2,4) satisfy the inequality.

Thus, the interval of x is (2,4).

(c)

To determine

### To solve: The inequality x(x−1)(x+2)>0.

Expert Solution

The solution of interval of x is (2,0)(1,).

### Explanation of Solution

Consider the inequality x(x1)(x+2)>0.

The inequality x(x1)(x+2)>0 will change sign at the critical values x=0,x=1andx=2.

The corresponding possible intervals of solution are (,2),(2,0),(0,1)and(1,).

It is clear that, the both intervals (2,0)and(1,) satisfy the inequality.

Thus, the solution of interval of x is (2,0)(1,).

(d)

To determine

### To solve: The inequality |x−4|<3.

Expert Solution

The solution of interval of x is (1,7).

### Explanation of Solution

Definition used:

The modulus function is to get the absolute numerical value of a number, irrespective of its positive or negative prefix sign.

Suppose the modulus function is |x|<a.

That is, a<x<a.

Calculation:

Consider the inequality |x4|<3.

Use the above mentioned definition and simplify the inequality as follows,

|x4|<33<x4<33+4<x<3+41<x<7

Thus, the solution of interval of x is (1,7).

(e)

To determine

### To solve: The inequality 2x−3x+1≤1.

Expert Solution

The solution of interval of x is (1,4].

### Explanation of Solution

Consider the inequality 2x3x+11.

Simplify the above inequality as follows,

2x3x+112x3x+1102x3x+1x+1x+102x3x1x+10

x4x+10

The inequality x4x+10 will change sign at the critical values x=4andx=1.

The corresponding possible intervals of solution are (,1),(1,4]and[4,).

It is clear that, the interval (1,4] satisfy the inequality.

Thus, the solution of interval of x is (1,4].

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