Math / Algebra

Question(b) By writing $|x| \geq a$ in the equivalent for $x \geq a$ or $x \leq-a$ , determine the solution set for $\left|\frac{x-3}{2 x-1}\right| \geq \frac{1}{2}$ [6 marks]

Studdy Solution

STEP 1

1. The inequality involves an absolute value, which can be split into two separate inequalities.
2. The expression inside the absolute value is a rational function.
3. The inequality $|x| \geq a$ is equivalent to $x \geq a$ or $x \leq -a$ .

STEP 2

1. Rewrite the inequality without the absolute value.
2. Solve the inequality $\frac{x-3}{2x-1} \geq \frac{1}{2}$ .
3. Solve the inequality $\frac{x-3}{2x-1} \leq -\frac{1}{2}$ .
4. Combine the solution sets from steps 2 and 3.
5. Consider any restrictions on the domain of the function.

STEP 3

Rewrite the inequality $\left|\frac{x-3}{2x-1}\right| \geq \frac{1}{2}$ as two separate inequalities:
$\frac{x-3}{2x-1} \geq \frac{1}{2} \quad \text{or} \quad \frac{x-3}{2x-1} \leq -\frac{1}{2}$

STEP 4

Solve the inequality $\frac{x-3}{2x-1} \geq \frac{1}{2}$ :
First, cross-multiply to eliminate the fraction, being mindful of the sign of $2x-1$ :
$2(x-3) \geq (2x-1)$
Simplify and solve for $x$ :
$2x - 6 \geq 2x - 1$
Subtract $2x$ from both sides:
$-6 \geq -1$
This inequality is always false, so there is no solution from this inequality.

STEP 5

Solve the inequality $\frac{x-3}{2x-1} \leq -\frac{1}{2}$ :
Cross-multiply to eliminate the fraction:
$2(x-3) \leq -(2x-1)$
Simplify and solve for $x$ :
$2x - 6 \leq -2x + 1$
Add $2x$ to both sides:
$4x - 6 \leq 1$
Add 6 to both sides:
$4x \leq 7$
Divide by 4:
$x \leq \frac{7}{4}$

STEP 6

Combine the solution sets from steps 2 and 3:
Since the first inequality had no solution, the solution set is determined solely by the second inequality:
$x \leq \frac{7}{4}$

STEP 7

Consider any restrictions on the domain of the function:
The expression $\frac{x-3}{2x-1}$ is undefined when the denominator is zero, i.e., $2x - 1 = 0$ , which gives $x = \frac{1}{2}$ .
Therefore, the solution set must exclude $x = \frac{1}{2}$ .
The final solution set is:
$x \leq \frac{7}{4}, \, x \neq \frac{1}{2}$
The solution set is:
$\boxed{x \leq \frac{7}{4}, \, x \neq \frac{1}{2}}$

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Question(b) By writing $|x| \geq a$ in the equivalent for $x \geq a$ or $x \leq-a$ , determine the solution set for $\left|\frac{x-3}{2 x-1}\right| \geq \frac{1}{2}$ [6 marks]

Studdy Solution

STEP 1

1. The inequality involves an absolute value, which can be split into two separate inequalities.
2. The expression inside the absolute value is a rational function.
3. The inequality $|x| \geq a$ is equivalent to $x \geq a$ or $x \leq -a$ .

STEP 2

1. Rewrite the inequality without the absolute value.
2. Solve the inequality $\frac{x-3}{2x-1} \geq \frac{1}{2}$ .
3. Solve the inequality $\frac{x-3}{2x-1} \leq -\frac{1}{2}$ .
4. Combine the solution sets from steps 2 and 3.
5. Consider any restrictions on the domain of the function.

STEP 3

Rewrite the inequality $\left|\frac{x-3}{2x-1}\right| \geq \frac{1}{2}$ as two separate inequalities:
$\frac{x-3}{2x-1} \geq \frac{1}{2} \quad \text{or} \quad \frac{x-3}{2x-1} \leq -\frac{1}{2}$

STEP 4

STEP 5

Solve the inequality $\frac{x-3}{2x-1} \leq -\frac{1}{2}$ :
Cross-multiply to eliminate the fraction:
$2(x-3) \leq -(2x-1)$
Simplify and solve for $x$ :
$2x - 6 \leq -2x + 1$
Add $2x$ to both sides:
$4x - 6 \leq 1$
Add 6 to both sides:
$4x \leq 7$
Divide by 4:
$x \leq \frac{7}{4}$

STEP 6

Combine the solution sets from steps 2 and 3:
Since the first inequality had no solution, the solution set is determined solely by the second inequality:
$x \leq \frac{7}{4}$

STEP 7

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Question(b) By writing ∣x∣≥a|x| \geq a∣x∣≥a in the equivalent for x≥ax \geq ax≥a or x≤−ax \leq-ax≤−a, determine the solution set for ∣x−32x−1∣≥12\left|\frac{x-3}{2 x-1}\right| \geq \frac{1}{2}∣∣​2x−1x−3​∣∣​≥21​ [6 marks]

Studdy Solution

STEP 1

1. The inequality involves an absolute value, which can be split into two separate inequalities.2. The expression inside the absolute value is a rational function.3. The inequality ∣x∣≥a |x| \geq a ∣x∣≥a is equivalent to x≥a x \geq a x≥a or x≤−a x \leq -a x≤−a.

STEP 2

STEP 3

STEP 4

STEP 5

STEP 6

Combine the solution sets from steps 2 and 3:Since the first inequality had no solution, the solution set is determined solely by the second inequality:x≤74 x \leq \frac{7}{4} x≤47​

STEP 7

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Question(b) By writing $|x| \geq a$ in the equivalent for $x \geq a$ or $x \leq-a$ , determine the solution set for $\left|\frac{x-3}{2 x-1}\right| \geq \frac{1}{2}$ [6 marks]

1. The inequality involves an absolute value, which can be split into two separate inequalities.
2. The expression inside the absolute value is a rational function.
3. The inequality $|x| \geq a$ is equivalent to $x \geq a$ or $x \leq -a$ .

Combine the solution sets from steps 2 and 3:
Since the first inequality had no solution, the solution set is determined solely by the second inequality:
$x \leq \frac{7}{4}$