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Chapter 11: Problem 1

If \(x>0,\) what is the value of \(x ?\) (1) \(x>5\) (2) \(40-x^{2}=4\)

Short Answer

Expert verified
The value of \(x\) is 6.

Step by step solution

01

- Analyze Given Statements

First, examine statement (1): given that \(x > 5\). This tells us that \(x\) is greater than 5 but does not specify a single value for \(x\).
02

- Solve Statement (2) Equation

Next, use statement (2): \(40 - x^2 = 4\). Rearrange this equation to isolate \(x^2\): \(40 - 4 = x^2\). Simplifying, we get: \(36 = x^2\). Finally, we take the square root of both sides: \(x = \pm6\).
03

- Apply Condition from Statement (1)

From statement (2), \(x\) can be \(6\) or \(-6\). Given the condition \(x > 0\) and from statement (1): \(x > 5\), the only possible value is \(x = 6\).

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Key Concepts

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

GMAT math strategies
When tackling GMAT math problems, efficient problem-solving and strategy are key. For the given exercise, we need to determine the value of \( x \) using the two provided statements. Start by analyzing each statement independently and then combine the information. GMAT problems often use conditions that narrow down possibilities, helping to find a unique solution. Remember these points:

  • Read each statement carefully and determine what information it provides.
  • Isolate and simplify any equations to make solving easier.
  • Combine the information from all statements to check for a consistent solution that meets all conditions.
With practice, you can develop a systematic approach to handle even the most complex problem efficiently!
solving quadratic equations
Quadratic equations are a central topic in algebra, essential for the GMAT and beyond. A standard quadratic equation takes the form: \( ax^2 + bx + c = 0 \). To solve these equations, utilize the following steps:

  • First, rearrange and simplify the equation if necessary. For instance, from statement (2), we started with \( 40 - x^2 = 4 \).
  • Next, isolate the term involving \( x^2 \). Subtract 4 from both sides to get \( 40 - 4 = x^2 \) which simplifies to \( 36 = x^2 \).
  • The solution to \( x^2 = 36 \) is \( x = \pm 6 \). Here, both positive and negative values are solutions because squaring either will result in a positive value.
Solving quadratic equations often gives us two potential solutions, which we then need to verify against any given conditions.
inequalities in algebra
Inequalities add another layer of complexity to algebra problems. In the given exercise, combining inequalities with the solutions of a quadratic equation is necessary. Here鈥檚 how you can manage them:

  • Understand the inequality: For the condition \( x > 0 \) given in the problem, we are specifically looking for positive solutions.
  • Combine inequalities: When we add the information that \( x > 5 \) from statement (1), it further narrows down the possible values of \( x \).
  • Cross-check solutions: From \( x^2 = 36 \), we get \( x = 6 \) or \( x = -6 \). Since the inequalities \( x > 0 \) and \( x > 5 \) apply, only \( x = 6 \) satisfies both conditions.
Handling inequalities requires careful observation and consistent application of given conditions to eliminate non-relevant solutions. Always verify your solution meets all criteria set by the problem.

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Most popular questions from this chapter

Team \(\mathrm{X}\) won 40 basketball games. What percent of its basketball games did Team \(\mathrm{X}\) win? (1) Team X played the same number of basketball games as Team Y. (2) Team Y won 45 games, representing 2.5 percent of the basketball games it played.

If \(x\) and \(y\) are prime numbers, is \(y(x-3)\) odd? (1) \(x>10\) (2) \(y<3\)

If \(2 b-a^{2}=18,\) what is the value of \(b ?\) (1) \(a^{2}=1,156\) (2) \(a>0\)

If \(y>0,\) is \(x\) less than \(0 ?\) (1) \(x y=16\) (2) \(x-y=6\)

If \(z\) is an integer, what is the units digit of \(z^{3} ?\) (1) \(z\) is a multiple of 5 (2) \(\sqrt{z}\) is an integer.
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