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Chapter 0: Problem 32

Solve each linear inequality. $$-5 x \leq 30$$

Short Answer

Expert verified
The solution to the inequality \( -5x \leq 30 \) is \( x \geq -6 \).

Step by step solution

01

Identify the Inequality

Given linear inequality is \( -5x \leq 30 \). The goal is to solve for x.
02

Divide by the Coefficient of x

Divide both sides of the equality by the negative coefficient of x, which is -5. Remember that the direction of the inequality changes when dividing by a negative number.\n So, \( x \geq -6 \) is the solution.

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Key Concepts

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

Linear Inequality
A linear inequality is much like a linear equation, except it shows a relationship where one side is not necessarily equal to the other, but rather greater than, less than, or possibly equal to. In simple terms, it looks for a range of values that satisfy the condition instead of a single number.

For instance, the inequality \( -5x \leq 30 \) states that when we multiply \( -5 \) by \( x \) the result should be less than or equal to 30. This kind of inequality is fundamental in mathematics as it helps define intervals within which solutions are valid, shaping our understanding of relationship and order between quantities.
Dividing by a Negative
When dividing both sides of an inequality by a negative number, the direction of the inequality sign must be flipped. This is a unique property that distinguishes inequalities from equations. An easy way to remember this is: if you're making the inequality 'colder' by dividing by a negative, the inequality 'shivers' and flips the other way.

For example, our original inequality, \( -5x \leq 30 \) demands this rule when we divide both sides by \( -5 \) to isolate \( x \). As a result, we get \( x \geq -6 \) instead of the anticipated \( x \leq -6 \) if we were dealing with a typical equation.
Inequality Solution
The solution to an inequality is the set of all values that satisfy the condition it stipulates. Unlike an equation that often has just one or two solutions, an inequality usually has an infinite number of solutions, forming a range.

After solving \( -5x \leq 30 \) and reversing the inequality to \( x \geq -6 \) when we divided by a negative number, we have determined the solution. It tells us that \( x \) can be any number greater than or equal to \( -6 \) 鈥� this encompasses a vast set of numbers not limited to integers. Understanding that these solutions comprise an entire interval on the number line is crucial.
Coefficient
A coefficient in the context of linear inequalities is the number that multiplies the variable. In the inequality \( -5x \leq 30 \) the coefficient of \( x \) is \( -5 \). It has a significant role: not only does it affect the value of \( x \) needed to make the inequality true, but it also determines the behavior of inequality when we perform operations like dividing.

Understanding that the coefficient controls the 'steepness' and direction of the inequality's graph on a coordinate plane can help visualize these solutions. Getting comfortable with how a coefficient functions will sharpen your problem-solving skills for a variety of math problems.

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Most popular questions from this chapter

Factor completely. $$(x-5)^{-\frac{1}{2}}(x+5)^{-\frac{1}{2}}-(x+5)^{\frac{1}{2}}(x-5)^{-\frac{5}{2}}$$

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. An elevator at a construction site has a maximum capacity of 3000 pounds. If the elevator operator weighs 245 pounds and each cement bag weighs 95 pounds, how many bags of cement can be safely lifted on the elevator in one trip?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. In an inequality such as \(5 x+4<8 x-5,\) I can avoid division by a negative number depending on which side I collect the variable terms and on which side I collect the constant terms.

Solve each equation. $$\left|x^{2}+6 x+1\right|=8$$

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A company manufactures and sells personalized stationery. The weekly fixed cost is \(\$ 3000\) and it costs \(\$ 3.00\) to produce each package of stationery. The selling price is \(\$ 5.50\) per package. How many packages of stationery must be produced and sold each week for the company to generate a profit?
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