Problem 5 Determine whether each value of ... [FREE SOLUTION]

Chapter 2: Problem 5

Determine whether each value of $x$ is a solution of the inequality. $$x^{2}-3<0$$ (a) $x=3$ (b) $x=0$ (c) $x=\frac{3}{2}$ (d) $x=-5$

Short Answer

Expert verified

Out of the given numbers, $x=0$ and $x=\frac{3}{2}$ are solutions to the inequality $x^2 - 3 < 0$.

Step by step solution

Substitute $x=3$

Substitute $x=3$ into the inequality giving: $3^2-3<0$ which simplifies to $9-3<0$, leading to $6<0$. This is not true, so $x=3$ is not a solution.

Substitute $x=0$

Substitute $x=0$ into the inequality giving: $0^2-3<0$ which simplifies to $-3<0$. This is true, so $x=0$ is a solution.

Substitute $x=\frac{3}{2}$

Substitute $x=\frac{3}{2}$ into the inequality providing: $(\frac{3}{2})^2-3<0$ which simplifies to $\frac{9}{4}-3<0$ leading to $\frac{-3}{4} <0$. This inequality is true, so $x=\frac{3}{2}$ is a solution.

Substitute $x=-5$

Substitute $x=-5$ into the inequality yielding: $(-5)^2-3<0$ which simplifies to $25-3<0$, leading to $22<0$. This is not correct, so $x=-5$ is not a solution.

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Key Concepts

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

Quadratic Inequality

A quadratic inequality is similar to a quadratic equation, but instead of an equal sign, it includes inequality symbols such as <, >, 鈮�, or 鈮�. These inequalities often deal with expressions that have a variable raised to the second power, typically looking like this:

Quadratic: A mathematical expression where the highest power of the variable is 2, like in $x^2 - 3 < 0$.

When solving a quadratic inequality, the goal is to find the range of values for the variable that makes the inequality true. Remember, unlike equations, inequalities represent a spectrum of answers. For a quadratic inequality like $x^2 - 3 < 0$, the solution involves analyzing the parabolic graph of the quadratic equation and finding intervals where the inequality holds.

Substitution Method

The substitution method is a strategic approach used for solving equations or inequalities by testing specific values. It's a practical way to verify whether a particular value satisfies the given inequality. Here's how it works:

Take each potential solution and substitute it for the variable.
Simplify the expression to see if the inequality holds true.

Using this method makes it clear which values are solutions by directly testing them one by one. For example, in our problem, substituting values like $ x = 0 $ revealed that it satisfies $ x^2 - 3 < 0$, making it a solution. This approach is particularly helpful for straightforward verification with small or simple numbers.

Solution Verification

Verifying a solution means ensuring the values you've identified actually satisfy the inequality. After you substitute a value, you simplify the resulting expression to check the truthfulness of the inequality. Let's consider the steps:

After substituting a value into the inequality, simplify and see if the inequality statement is true or false.
If true, you've found a solution; if false, the value is not a solution.

This process of solution verification not only confirms the validity of your solution but also enhances understanding of how inequalities work. In our exercise, substituting $x=3$ gives $6 < 0$, which is false, so $x=3$ is not a solution.

Inequality Solving

Solving inequalities involves determining the set of values that satisfy the inequality condition. Unlike equalities, which often have one solution, inequalities generally have a range of solutions because they define a region on the number line. To solve inequalities:

Identify the quadratic expression and rearrange terms if needed.
Simplify the expression, substituting values, as needed, to check potential solutions.
Consider different approaches like graphing or algebraic methods to determine where the inequality holds true.

An important aspect of inequality solving is considering boundary points and testing values in different intervals. This ensures a comprehensive understanding of the overall behavior and solutions for the inequality at hand. In our case, testing values like $x = \frac{3}{2}$ showed that it satisfies $x^2 - 3 < 0$, indicating it is within the solution range.

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91影视

Determine whether each value of \(x\) is a solution of the inequality. $$x^{2}-3<0$$ (a) \(x=3\) (b) \(x=0\) (c) \(x=\frac{3}{2}\) (d) \(x=-5\)

Short Answer

Step by step solution

Substitute \(x=3\)

Substitute \(x=0\)

Substitute \(x=\frac{3}{2}\)

Substitute \(x=-5\)

Key Concepts

Quadratic Inequality

Substitution Method

Solution Verification

Inequality Solving

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