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Chapter 1: Problem 129

Challenge Problem Solve: \(x-4<2 x-3 \leq \frac{x+5}{3}\)

Short Answer

Expert verified
-1 < x \leq \frac{14}{5}

Step by step solution

01

Split the compound inequality

The compound inequality can be split into two separate inequalities: 1. Solve for the left inequality: \(x - 4 < 2x - 3\)2. Solve for the right inequality: \(2x - 3 \leq \frac{x + 5}{3}\)
02

Solve the left inequality

To solve \(x - 4 < 2x - 3\), follow these steps:1. Subtract \(x\) from both sides: \(-4 < x - 3\)2. Add 3 to both sides: \(-1 < x\) or \(x > -1\).
03

Solve the right inequality

To solve \(2x - 3 \leq \frac{x + 5}{3}\), follow these steps:1. Multiply both sides by 3 to eliminate the fraction: \(3(2x - 3) \leq x + 5\)2. Distribute the 3: \(6x - 9 \leq x + 5\)3. Subtract \(x\) from both sides: \(5x - 9 \leq 5\)4. Add 9 to both sides: \(5x \leq 14\)5. Divide by 5: \(x \leq \frac{14}{5}\).
04

Combine the solutions

From Step 2, we have \(x > -1\). From Step 3, we have \(x \leq \frac{14}{5}\). Combining these, the final solution is: \(-1 < x \leq \frac{14}{5}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inequality Operations
When working with inequalities, the goal is similar to solving equations: isolate the variable on one side.However, there are a few additional rules to keep in mind.
For instance, if you multiply or divide both sides of an inequality by a negative number, the inequality sign must flip direction. This does not happen when working with equalities. The basic operations, like adding, subtracting, multiplying, or dividing both sides of the inequality by the same non-zero number, stay consistent.
Carefully managing these operations ensures that the inequality remains valid.
Solving Linear Inequalities
Solving linear inequalities involves a sequence of steps aimed at isolating the variable. Start by simplifying the inequality through basic algebraic operations:
  • Perform addition or subtraction first to move constants to the other side.
  • Use multiplication or division to isolate the variable.

For example, with the left inequality from the exercise, we had:
\(x - 4 < 2x - 3\).
1. Subtracting \(x\) from both sides results in \(-4 < x - 3\).
2. Then, by adding 3 to both sides we get \(-1 < x\), or equivalently \(x > -1\).
This sequence ensures that we isolate \(x\) effectively, giving the valid solution. Ensuring accuracy in these steps is crucial.
Fraction Elimination
Fractions in inequalities can be tricky. The key strategy is to eliminate the fraction early. Multiply all terms by the denominator of the fraction.
For the right inequality from the exercise:
\(2x - 3 \frac{x + 5}{3}\)
we start by multiplying every term by 3.
This gives us:
\(3(2x - 3) \ \ x + 5\),
which simplifies to:
\(6x - 9 \ \ x + 5\).
With the fraction out of the way, we can proceed with solving as we did with simpler linear inequalities.
This big first step simplifies our calculations and keeps the inequality easier to manage.
Inequality Combination
After solving separate inequalities, the next step is to combine them. Combining retains the valid ranges for the variable from each part.
From the first inequality, we had:
\(x > -1\).
From the second inequality, we had:
\(x \ 14 \ 5\).
Combining these gives us the range that satisfies both conditions:
\(-1 < x \ 14 \ 5\).
This represents the intersection of the two solution sets and provides a clear, consolidated range for \(x\).
Ensuring clarity in this final combination step guarantees the solution is accurate and all-encompassing.

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Most popular questions from this chapter

The distance to the surface of the water in a well can sometimes be found by dropping an object into the well and measuring the time elapsed until a sound is heard. If \(t_{1}\) is the time (measured in seconds) that it takes for the object to strike the water, then \(t_{1}\) will obey the equation \(s=16 t_{1}^{2}\), where \(s\) is the distance (measured in feet). It follows that \(t_{1}=\frac{\sqrt{s}}{4}\). Suppose that \(t_{2}\) is the time that it takes for the sound of the impact to reach your ears. Because sound waves are known to travel at a speed of approximately 1100 feet per second, the time \(t_{2}\) to travel the distance \(s\) will be \(t_{2}=\frac{s}{1100} .\) See the illustration. Now \(t_{1}+t_{2}\) is the total time that elapses from the moment that the object is dropped to the moment that a sound is heard. We have the equation $$ \text { Total time elapsed }=\frac{\sqrt{s}}{4}+\frac{s}{1100} $$ Find the distance to the water's surface if the total time elapsed from dropping a rock to hearing it hit water is 4 seconds.

Physics A ball is thrown vertically upward from the top of a building 96 feet tall with an initial velocity of 80 feet per second. The distance \(s\) (in feet) of the ball from the ground after \(t\) seconds is \(s=96+80 t-16 t^{2}\). (a) After how many seconds does the ball strike the ground? (b) After how many seconds will the ball pass the top of the building on its way down?

Mixing Water and Antifreeze The cooling system of a certain foreign-made car has a capacity of 15 liters. If the system is filled with a mixture that is \(40 \%\) antifreeze, how much of this mixture should be drained and replaced by pure antifreeze so that the system is filled with a solution that is \(60 \%\) antifreeze?

Find the real solutions of each equation. Use a calculator to express any solutions rounded to two decimal places. $$ \pi(1+t)^{2}=\pi+1+t $$

Computing Grades In your Economics 101 class, you have scores of \(68,82,87,\) and 89 on the first four of five tests. To get a grade of B, the average of the first five test scores must be greater than or equal to 80 and less than 90 . (a) Solve an inequality to find the least score you can get on the last test and still earn a \(\mathrm{B}\). (b) What score do you need if the fifth test counts double?
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