Question

what is a counterexample to this claim? if x is any real number, then ( x^2 geq 1 ). select the correct answer. ( circ x = -2.5 ) ( circ x = -1.5 ) ( circ x = 0.5 ) ( circ x = 1.5 )

Explanation:

Step1: Recall counterexample definition

A counterexample is a value that makes the claim false. The claim is \( x^2 \geq 1 \) for all real \( x \). We need to find \( x \) where \( x^2 < 1 \).

Step2: Calculate \( x^2 \) for each option

• For \( x = -2.5 \): \( (-2.5)^2 = 6.25 \), \( 6.25 \geq 1 \) (does not contradict the claim).
• For \( x = -1.5 \): \( (-1.5)^2 = 2.25 \), \( 2.25 \geq 1 \) (does not contradict the claim).
• For \( x = 0.5 \): \( (0.5)^2 = 0.25 \), \( 0.25 < 1 \) (contradicts the claim, so this is a counterexample).
• For \( x = 1.5 \): \( (1.5)^2 = 2.25 \), \( 2.25 \geq 1 \) (does not contradict the claim).

Answer:

\( x = 0.5 \) (the option with \( x = 0.5 \))