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Solve for \( x \) : CLASSWORK \( \begin{array}{ll}\text { (a) } 3 x+4<5 \\ \text { (d) } 2(x-6)>0 & \text { (2) } \\ \text { (d) } \frac{x}{3} \geq \frac{x-1}{4} & \text { (3) }\end{array} \)

Ask by Frazier Gray. in South Africa
Mar 09,2025

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Answer

\( x < \frac{1}{3} \), \( x > 6 \), \( x \geq -3 \)

Solution

Solve the equation \( 3x+4<5 \). Solve the inequality by following steps: - step0: Solve for \(x\): \(3x+4<5\) - step1: Move the constant to the right side: \(3x<5-4\) - step2: Subtract the numbers: \(3x<1\) - step3: Divide both sides: \(\frac{3x}{3}<\frac{1}{3}\) - step4: Divide the numbers: \(x<\frac{1}{3}\) Solve the equation \( 2(x-6)>0 \). Solve the inequality by following steps: - step0: Solve for \(x\): \(2\left(x-6\right)>0\) - step1: Rewrite the expression: \(x-6>0\) - step2: Move the constant to the right side: \(x>0+6\) - step3: Remove 0: \(x>6\) Solve the equation \( \frac{x}{3} \geq \frac{x-1}{4} \). Solve the inequality by following steps: - step0: Solve for \(x\): \(\frac{x}{3}\geq \frac{x-1}{4}\) - step1: Multiply both sides: \(\frac{x}{3}\times 12\geq \frac{x-1}{4}\times 12\) - step2: Multiply the terms: \(4x\geq 3x-3\) - step3: Move the variable to the left side: \(4x-3x\geq -3\) - step4: Subtract the terms: \(x\geq -3\) The solutions to the given inequalities are: (a) \( x < \frac{1}{3} \) (d) \( x > 6 \) (d) \( x \geq -3 \)

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Let's solve the inequalities one by one! For the first inequality \(3x + 4 < 5\): 1. Subtract 4 from both sides: \(3x < 1\) 2. Divide by 3: \(x < \frac{1}{3}\) Now, for the second inequality \(2(x-6) > 0\): 1. Expand it: \(2x - 12 > 0\) 2. Add 12 to both sides: \(2x > 12\) 3. Divide by 2: \(x > 6\) Next, let's solve \(\frac{x}{3} \geq \frac{x-1}{4}\): 1. Cross multiply to eliminate the fractions: \(4x \geq 3(x - 1)\) 2. Expand: \(4x \geq 3x - 3\) 3. Subtract \(3x\) from both sides: \(x \geq -3\) So, we have the solution sets: 1. \(x < \frac{1}{3}\) 2. \(x > 6\) 3. \(x \geq -3\) These inequalities give you a range of values for \(x\) depending on which one you're interested in!

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