Example Question - inequality solution steps
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Solving Inequalities with Variable Isolation
To solve the inequality \(5g + 3 \leq 37\), we need to isolate the variable \(g\). Here are the steps: 1. Subtract 3 from both sides: \[5g + 3 - 3 \leq 37 - 3\] \[5g \leq 34\] 2. Divide both sides by 5: \[\frac{5g}{5} \leq \frac{34}{5}\] \[g \leq \frac{34}{5}\] \[g \leq 6.8\] Now, let's look at the answer choices to find which ones are less than or equal to 6.8: - \(g = -8\) is less than 6.8; hence it's a valid solution. - \(g = -5\) is less than 6.8; hence it's a valid solution. - \(g = -2\) is less than 6.8; hence it's a valid solution. - \(g = -1\) is less than 6.8; hence it's a valid solution. All of the given choices \(g = -8, g = -5, g = -2, g = -1\) are solutions to the inequality \(5g + 3 \leq 37\).
Solving Absolute Value Inequalities
To solve the inequality \( 9 - 8 | r + 5 | > -11 \), we must isolate the absolute value expression on one side of the inequality. Here are the steps to solve the inequality: 1. Add or subtract quantities from both sides of the inequality to move constant terms away from the absolute value expression: \[ 9 - 8 | r + 5 | + 11 > -11 + 11 \] \[ 20 - 8 | r + 5 | > 0 \] 2. Now we need to isolate the absolute value expression. To do this, we can subtract 20 from both sides: \[ 20 - 8 | r + 5 | - 20 > 0 - 20 \] \[ -8 | r + 5 | > -20 \] 3. To get rid of the coefficient in front of the absolute value, we divide both sides by -8. Remember that when you divide by a negative number, you have to flip the inequality sign: \[ \frac{-8 | r + 5 |}{-8} < \frac{-20}{-8} \] \[ | r + 5 | < \frac{20}{8} \] 4. Simplify the fraction: \[ | r + 5 | < 2.5 \] This inequality says that the distance from the point \( r + 5 \) to 0 on the number line is less than 2.5. We can break this down into two separate inequalities: If \( r + 5 \) is positive: \[ r + 5 < 2.5 \] If \( r + 5 \) is negative: \[ r + 5 > -2.5 \] Now solve for \( r \) in each inequality: For the case where \( r + 5 \) is positive: \[ r < 2.5 - 5 \] \[ r < -2.5 \] For the case where \( r + 5 \) is negative: \[ r > -2.5 - 5 \] \[ r > -7.5 \] So the solution to the inequality is: \[ -7.5 < r < -2.5 \] This means that \( r \) is greater than -7.5 and less than -2.5.
Solving an Inequality Involving Absolute Value
To solve the inequality \( 8 - 7|1 - 6s| < -6 \), we could follow these steps: 1. First, let's simplify the inequality by removing the constant term '8' from both sides: \[ 8 - 7|1 - 6s| - 8 < -6 - 8 \] \[ -7|1 - 6s| < -14 \] 2. Now, we divide both sides by '-7'. Remember that dividing by a negative flips the inequality sign: \[ \frac{-7|1 - 6s|}{-7} > \frac{-14}{-7} \] \[ |1 - 6s| > 2 \] 3. The inequality \( |1 - 6s| > 2 \) means that the expression inside the absolute value is either greater than 2 or less than -2. We can express this as two separate inequalities: \[ 1 - 6s > 2 \quad \text{or} \quad 1 - 6s < -2 \] 4. To solve \( 1 - 6s > 2 \), subtract '1' from both sides and then divide by '-6': \[ -6s > 1 \] \[ s < -\frac{1}{6} \] 5. To solve \( 1 - 6s < -2 \), subtract '1' from both sides and then divide by '-6': \[ -6s < -3 \] \[ s > \frac{1}{2} \] The solution to the inequality is therefore \( s < -\frac{1}{6} \) or \( s > \frac{1}{2} \).
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