Help, please! x^3 > x.........?

Question

Directions: Solve the inequality in terms of intervals and illustrate the sollution set on the real number line.

1.) x^3 > x

2.) x^2 < 3

3.) 1/x < 4

4.) x^2 + x > 1

Answer 1

1st problem:

x³ &gt; x

x² &gt; 1

x &gt; 1

2nd problem:

x² &lt; 3

x &lt; √3

3rd problem:

1/x &lt; 4

1 &lt; 4x

1/4 &lt; x

x &gt; 1/4

4th problem:

x² + x &gt; 1

x² + 1/2x &gt; 1 + (1/2)²

x² + 1/2x &gt; 4/4 + 1/4

(x + 1/2)² &gt; 5/4

x + 1/2 &gt; 1.118034

x &gt; 0.618034 1st inequality

x &gt; - 1.118034 - 0.5

x &gt; - 1.618034 2nd inequality

Answer 2

q1

x^3 &gt; x

x^3 - x &gt; 0

x(x^2 - 1) &gt; 0

x(x - 1)(x + 1) &gt; 0

answer : -1 &lt; x &lt; 0, x &gt; 1

q2

x^2 &lt; 3

x^2 - 3 &lt; 0

(x + √3)(x - √3) &lt; 0

answer : -√3 &lt; x &lt; √3

q3

1/x &lt; 4

1/x - 4 &lt; 0

(1 - 4x) / x &lt; 0

1 - 4x &lt; 0 AND x &gt; 0, or 1 - 4x &gt; 0 AND x &lt; 0

answer : x &lt; 0, x &gt; 1/4

q4

x^2 + x &gt; 1

x^2 + x + 1/4 &gt; 1 + 1/4

(x + 1/2)^2 &gt; 5/4

x + 1/2 &lt; -√5 /2, x + 1/2 &gt; √5 /2

answer : x &lt; (-1 - √5) /2, x &gt; (-1 + √5) /2