Question

solving inequaliti
directions: solve each inequality. circle
corresponding areas on the coloring she
1 $-90\geq18x$
2 $\frac{x}{3}>-8$
3 $x - 10 < -26$
4 $19 - x\geq21$
5 $\frac{x}{3}+7>12$
6 $\frac{-5 + x}{4}>1$
7 $-9(4 + x)>-126$
8 $9(x - 7)<-90$
9 $9\geq\frac{1 + x}{2}$
10 $-23<1 - 4x - 8$
11 $0\geq6x - 6x$
12 $15>6x + 1 - 4$
13 $-96\leq8(x - 6)$
14 $1 - 4(7 - 6x)<141$
15 $-134<8x - 5(1 - 7x)$
16 $-6 + 8(6x + 2)<7x + 10$
17 $-3 + 3x\leq x + 2(x - 4)$
18 $-22 - 4x<-(4 - 5x)$

Explanation:

Step1: Solve inequality 1

Divide both sides of \(-90\geq18x\) by \(18\).
\(\frac{-90}{18}\geq\frac{18x}{18}\), so \(x\leq - 5\).

Step2: Solve inequality 2

Multiply both sides of \(\frac{x}{3}>-8\) by \(3\).
\(x>-24\).

Step3: Solve inequality 3

Add \(10\) to both sides of \(x - 10<-26\).
\(x<-26 + 10\), so \(x<-16\).

Step4: Solve inequality 4

Subtract \(19\) from both sides of \(19-x\geq21\): \(-x\geq21 - 19\), then \(-x\geq2\). Multiply both sides by \(-1\) (and reverse the inequality sign), so \(x\leq - 2\).

Step5: Solve inequality 5

Subtract \(7\) from both sides of \(\frac{x}{3}+7>12\): \(\frac{x}{3}>12 - 7\), \(\frac{x}{3}>5\). Multiply both sides by \(3\), so \(x>15\).

Step6: Solve inequality 6

Multiply both sides of \(\frac{-5 + x}{4}>1\) by \(4\): \(-5+x>4\). Add \(5\) to both sides, so \(x>9\).

Step7: Solve inequality 7

First, distribute \(-9\) on the left - hand side: \(-36-9x>-126\). Add \(36\) to both sides: \(-9x>-126 + 36\), \(-9x>-90\). Divide both sides by \(-9\) (and reverse the inequality sign), so \(x < 10\).

Step8: Solve inequality 8

First, distribute \(9\) on the left - hand side: \(9x-63<-90\). Add \(63\) to both sides: \(9x<-90 + 63\), \(9x<-27\). Divide both sides by \(9\), so \(x<-3\).

Step9: Solve inequality 9

Multiply both sides of \(9\geq\frac{1 + x}{2}\) by \(2\): \(18\geq1 + x\). Subtract \(1\) from both sides, so \(x\leq17\).

Step10: Solve inequality 10

Simplify the right - hand side: \(-23<1-4x - 8\), \(-23<-4x-7\). Add \(7\) to both sides: \(-23 + 7<-4x\), \(-16<-4x\). Divide both sides by \(-4\) (and reverse the inequality sign), so \(x < 4\).

Step11: Solve inequality 11

Simplify the right - hand side: \(0\geq6x-6x\), \(0\geq0\). This is True for all real \(x\).

Step12: Solve inequality 12

Simplify the right - hand side: \(15>6x+1 - 4\), \(15>6x-3\). Add \(3\) to both sides: \(15 + 3>6x\), \(18>6x\). Divide both sides by \(6\), so \(x < 3\).

Step13: Solve inequality 13

First, distribute \(8\) on the right - hand side: \(-96\leq8x-48\). Add \(48\) to both sides: \(-96 + 48\leq8x\), \(-48\leq8x\). Divide both sides by \(8\), so \(x\geq - 6\).

Step14: Solve inequality 14

First, distribute \(-4\) on the left - hand side: \(1-28 + 24x<141\), \(-27+24x<141\). Add \(27\) to both sides: \(24x<141 + 27\), \(24x<168\). Divide both sides by \(24\), so \(x < 7\).

Step15: Solve inequality 15

First, distribute \(-5\) on the right - hand side: \(-134<8x-5 + 35x\), \(-134<43x-5\). Add \(5\) to both sides: \(-134 + 5<43x\), \(-129<43x\). Divide both sides by \(43\), so \(x>-3\).

Step16: Solve inequality 16

First, distribute \(8\) on the left - hand side: \(-6 + 48x+16<7x + 10\), \(48x + 10<7x+10\). Subtract \(7x\) from both sides: \(41x+10<10\). Subtract \(10\) from both sides: \(41x<0\). Divide both sides by \(41\), so \(x < 0\).

Step17: Solve inequality 17

First, distribute \(2\) on the right - hand side: \(-3 + 3x\leq x+2x-8\), \(-3 + 3x\leq3x-8\). Subtract \(3x\) from both sides: \(-3\leq - 8\). This is False for all real \(x\).

Step18: Solve inequality 18

First, distribute the negative sign on the right - hand side: \(-22-4x<-4 + 5x\). Add \(4x\) to both sides: \(-22<-4 + 9x\). Add \(4\) to both sides: \(-18<9x\). Divide both sides by \(9\), so \(x>-2\).

Answer:

1. \(x\leq - 5\)
2. \(x>-24\)
3. \(x<-16\)
4. \(x\leq - 2\)
5. \(x>15\)
6. \(x>9\)
7. \(x < 10\)
8. \(x<-3\)
9. \(x\leq17\)
10. \(x < 4\)
11. All real \(x\)
12. \(x < 3\)
13. \(x\geq - 6\)
14. \(x < 7\)
15. \(x>-3\)
16. \(x < 0\)
17. No solution
18. \(x>-2\)