Question

triangle inequalities
review! solve the following inequalities. watch out for the flippers!

1. $5x - 18 > 2x + 3$
2. $8x + 7 > 10x - 15$
3. $9x - 26 > 14x - 40$
4. $9 - 2x > 57 - 10x$

directions: if the sides of a triangle have the given lengths, find a range of possible $x$-values. since you do not know which two sides are the shortest, you must account for all possibilities.

1. $ab = 4x + 25$, $bc = 3x - 2$, $ac = 9x - 5$

range of $x$-values:

1. $mn = x - 1$, $np = 9x - 68$, $mp = 5x - 4$

range of $x$-values:

Explanation:

Step1: Isolate x terms

$5x - 2x > 3 + 18$

Step2: Simplify both sides

$3x > 21$

Step3: Solve for x

$x > \frac{21}{3}$
$x > 7$

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Step1: Isolate x terms

$7 + 15 > 10x - 8x$

Step2: Simplify both sides

$22 > 2x$

Step3: Solve for x

$x < \frac{22}{2}$
$x < 11$

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Step1: Isolate x terms

$-26 + 40 > 14x - 9x$

Step2: Simplify both sides

$14 > 5x$

Step3: Solve for x

$x < \frac{14}{5}$
$x < 2.8$

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Step1: Isolate x terms

$-2x + 10x > 57 - 9$

Step2: Simplify both sides

$8x > 48$

Step3: Solve for x

$x > \frac{48}{8}$
$x > 6$

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Step1: Ensure sides are positive

$3x - 2 > 0 \implies x > \frac{2}{3}$
$4x + 25 > 0$ (always true for $x>\frac{2}{3}$)
$9x - 5 > 0$ (always true for $x>\frac{2}{3}$)

Step2: Apply triangle inequality 1

$(4x+25)+(3x-2) > 9x-5$
$7x + 23 > 9x - 5$
$28 > 2x \implies x < 14$

Step3: Apply triangle inequality 2

$(4x+25)+(9x-5) > 3x-2$
$13x + 20 > 3x - 2$
$10x > -22 \implies x > -2.2$ (irrelevant, since $x>\frac{2}{3}$)

Step4: Apply triangle inequality 3

$(3x-2)+(9x-5) > 4x+25$
$12x - 7 > 4x + 25$
$8x > 32 \implies x > 4$

Step5: Combine valid ranges

$\frac{2}{3} < x <14$ and $x>4$ gives $4 < x <14$

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Step1: Ensure sides are positive

$x - 1 > 0 \implies x > 1$
$9x - 68 > 0 \implies x > \frac{68}{9} \approx 7.56$
$5x - 4 > 0$ (always true for $x>\frac{68}{9}$)

Step2: Apply triangle inequality 1

$(x-1)+(9x-68) > 5x-4$
$10x - 69 > 5x - 4$
$5x > 65 \implies x > 13$

Step3: Apply triangle inequality 2

$(x-1)+(5x-4) > 9x-68$
$6x - 5 > 9x - 68$
$63 > 3x \implies x < 21$

Step4: Apply triangle inequality 3

$(9x-68)+(5x-4) > x-1$
$14x - 72 > x - 1$
$13x > 71 \implies x > \frac{71}{13} \approx 5.46$ (irrelevant, since $x>13$)

Step5: Combine valid ranges

$x>\frac{68}{9}$ and $13 < x <21$ gives $13 < x <21$

Answer:

1. $x > 7$
2. $x < 11$
3. $x < \frac{14}{5}$ (or $x < 2.8$)
4. $x > 6$
5. $4 < x < 14$
6. $13 < x < 21$