12.3: Exercises

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Exercise $$\PageIndex{1}$$

Solve for $$x$$.

1. $$5x+6\leq 21$$
2. $$3+4x> 10x$$
3. $$2x+8\geq 6x+24$$
4. $$9-3x< 2x-13$$
5. $$-5 \leq 2x+5\leq 19$$
6. $$15> 7-2x\geq 1$$
7. $$3x+4 \leq 6x-2\leq 8x+5$$
8. $$5x+2< 4x-18\leq 7x+11$$
1. $$x \leq 3$$
2. $$\dfrac{1}{2}>x$$
3. $$-4 \geq x$$
4. $$x>\dfrac{22}{5}$$
5. $$-5 \leq x \leq 7$$
6. $$-4<x \leq 3$$
7. $$x \geq 2$$ (this then also implies $$x \geq-\dfrac{7}{2}$$)
8. no solution

Exercise $$\PageIndex{2}$$

Solve for $$x$$.

1. $$x^2-5x-14>0$$
2. $$x^2-2x\geq 35$$
3. $$x^2-4\leq 0$$
4. $$x^2+3x-3<0$$
5. $$2x^2+2x\leq 12$$
6. $$3x^2<2x+1$$
7. $$x^2-4x+4>0$$
8. $$x^3-2x^2-5x+6\geq 0$$
9. $$x^3+4x^2+3x+12 <0$$
10. $$-x^3-4x<-4x^2$$
11. $$x^4-10x^2+9\leq 0$$
12. $$x^4-5x^3+5x^2+5x<6$$
13. $$x^4-5x^3+6x^2>0$$
14. $$x^5-6x^4+x^3+24x^2-20x\leq 0$$
15. $$x^5-15x^4+85x^3-225x^2+274x-120\geq 0$$
16. $$x^{11}-x^{10}+x-1\leq 0$$
1. $$(-\infty,-2) \cup(7, \infty)$$
2. $$(-\infty,-5] \cup[7, \infty)$$
3. $$[-2,2]$$
4. $$\left(\dfrac{-3-\sqrt{21}}{2}, \dfrac{-3+\sqrt{21}}{2}\right)$$
5. $$[-3,2]$$
6. $$\left(-\dfrac{1}{3}, 1\right)$$
7. $$\mathbb{R}-\{2\}$$
8. $$[-2,1] \cup[3, \infty)$$
9. $$(-\infty,-4)$$
10. $$(0,2) \cup(2, \infty)$$
11. $$[-3,-1] \cup[1,3]$$
12. $$(-1,1) \cup(2,3)$$
13. $$(-\infty, 0) \cup(0,2) \cup(3, \infty)$$
14. $$(-\infty,-2] \cup[0,1] \cup[2,5]$$
15. $$[1,2] \cup[3,4] \cup [5, \infty)$$
16. $$(-\infty, 1]$$

Exercise $$\PageIndex{3}$$

Find the domain of the functions below.

1. $$f(x)=\sqrt{x^2-8x+15}$$
2. $$f(x)=\sqrt{9x-x^3}$$
3. $$f(x)=\sqrt{(x-1)(4-x)}$$
4. $$f(x)=\sqrt{(x-2)(x-5)(x-6)}$$
5. $$f(x)=\dfrac{5}{\sqrt{6-2x}}$$
6. $$f(x)=\dfrac{1}{\sqrt{x^2-6x-7}}$$
1. $$D=(-\infty, 3] \cup[5, \infty)$$
2. $$D=(-\infty,-3] \cup[0,3]$$
3. $$D=[1,4]$$
4. $$D=[2,5] \cup[6, \infty)$$
5. $$D=(-\infty, 3)$$
6. $$D=(-\infty,-1) \cup(7, \infty)$$

Exercise $$\PageIndex{4}$$

Solve for $$x$$.

1. $$\dfrac{x-5}{2-x}>0$$
2. $$\dfrac{4x-4}{x^2-4}\geq 0$$
3. $$\dfrac{x-2}{x^2-4x-5}< 0$$
4. $$\dfrac{x^2-9}{x^2-4}\geq 0$$
5. $$\dfrac{x-3}{x+3}\leq 4$$
6. $$\dfrac{1}{x+10}> 5$$
7. $$\dfrac{2}{x-2}\leq \dfrac{5}{x+1}$$
8. $$\dfrac{x^2}{x+4}\leq x$$
1. $$(2,5)$$
2. $$(-2,1] \cup(2, \infty)$$
3. $$(-\infty,-1) \cup(2,5)$$
4. $$(-\infty,-3] \cup (-2,2) \cup[3, \infty)$$
5. $$(-\infty,-5] \cup(-3, \infty)$$
6. $$(-10,-9.8)$$
7. $$(-1,2) \cup [4, \infty)$$
8. $$(-\infty,-4) \cup[0, \infty)$$

Exercise $$\PageIndex{5}$$

Solve for $$x$$.

1. $$|2x+7|>9$$
2. $$|6x+2|<3$$
3. $$|5-3x|\geq 4$$
4. $$|-x-7|\leq 5$$
5. $$|1-8x|\geq 3$$
6. $$1>\left|2+\dfrac{x}{5}\right|$$
1. $$(-\infty,-8) \cup(1, \infty)$$
2. $$\left(\dfrac{-5}{6}, \dfrac{1}{6}\right)$$
3. $$\left(-\infty, \dfrac{1}{3}\right] \cup[3, \infty)$$
4. $$[-12,-2]$$
5. $$\left(-\infty,-\dfrac{1}{4}\right] \cup\left[\dfrac{1}{2}, \infty\right)$$
6. $$(-15,-5)$$