Unit Name: Polynomial and Rational Functions Solve the following inequality algebraically.

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Unit Name: Polynomial and Rational Functions Solve the following inequality algebraically.

Juan Hewlett 2021-12-11 Answered

Unit Name: Polynomial and Rational Functions
Solve the following inequality algebraically. Explain your process.

8 x - 3 \leq 2 x + 1 \leq 17 x - 8

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Karen Robbins

Answered 2021-12-12 Author has 49 answers

Split the inequality into two:

8 x - 3 \leq 2 x + 1 and 2 x + 1 \leq 17 x - 8

Isolate x from each inequality using inverse operations.

8 x - 3 \leq 2 x + 1

8 x - 2 x \leq 3 + 1

6 x \leq 4

x \leq \frac{2}{3}

AND

2 x + 1 \leq 17 x - 8

2 x - 17 x \leq - 8 - 1

- 15 x + 15 x \leq - 9 + 15 x

0 \leq - 9 + 15 x

\frac{3}{5} \leq x

Therefore, the solution is:

\frac{3}{5} \leq x \leq \frac{2}{3}

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Shannon Hodgkinson

Answered 2021-12-13 Author has 34 answers

Solve the following inequality algebraically.
Explain your process.

8 x - 3 \leq 2 x + 1 \leq 17 x - 8

We've

8 x - 3 \leq 2 x + 1 \leq 17 x - 8

. (1)

\Rightarrow 8 x - 3 - 1 \leq 2 x + 1 - 1 \leq 17 x - 8 - 1

\Rightarrow 8 x - 4 \leq 2 x \leq 17 x - 9

Now dividing above equation by 2 we get

\Rightarrow \frac{8 x - 4}{2} \leq \frac{2 x}{2} \leq \frac{17}{2} x - \frac{9}{2}

\Rightarrow 4 x - 2 \leq x \leq \frac{17}{2} x - \frac{9}{2}

(2)
Now from equation (2) we've two cases.
(1)

x \geq 4 x - 2

\Rightarrow

adding 2 to both sides we get

\Rightarrow x + 2 \geq 4 x - 2 + 2

\Rightarrow x + 2 \geq 4 x

Subtracting x from both sides

\Rightarrow x + 2 - x \geq 4 x - x

\Rightarrow 2 \geq 3 x

\Rightarrow x \leq \frac{2}{3}

x \leq \frac{17}{2} x - \frac{9}{2}

adding

\frac{9}{2}

to both sides we get

\Rightarrow x + \frac{9}{2} \leq \frac{17}{2} x - \frac{9}{2} + \frac{9}{2}

\Rightarrow x + \frac{9}{2} \leq \frac{17}{2} x

Subtracting x from both sides

\Rightarrow x + \frac{9}{2} - x \leq \frac{17}{2} x - x

\Rightarrow \frac{q}{2} \leq \frac{17 x - 2 x}{2}

\Rightarrow \frac{9}{2} \leq \frac{15 x}{2}

Multiplying 2 to both sides we get

Undefined control sequence \cancel

\Rightarrow 9 \leq 15 x

\Rightarrow x \geq \frac{3}{5}

(4)
From equation (3) and equation (4) we get

\frac{3}{5} \leq x \leq \frac{2}{3}

\Rightarrow 0.6 \leq x \leq 0.6666

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Unit Name: Polynomial and Rational Functions Solve the following inequality algebraically.

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